A Kinematic Self-Similar Rupture Process for Earthquakes

Bulletin of the Seismological Society of America, Vol. 84, No. 4, pp. 1216-1228, August 1994 A Kinematic Self-Similar Rupture Process for Earthquakes by A. Herrero and P. Bernard Abstract The basic assumption that the self-similarity and the spectral law of the seismic body-wave radiation (e.g., o)-square model) must find their origin in some simple self-similar process during the seismic rupture led us to construct a kinematic, self-similar model of earthquakes. It is first assumed that the amplitude of the slip distribution high-pass filtered at high wavenumber does not depend on the size of the ruptured fault. This leads to the following "k-square" model for the slip spectrum, for k > IlL: Ao- L ul(k) = C-- where L is the ruptured fault dimension, k the radial wavenumber, Ao- the mean stress drop, /x the rigidity, and C an adimensional constant of the order of 1. The associated stress-drop spectrum, for k > 1/L, is approximated by L = k The rupture front is assumed to propagate on the fault plane with a constant velocity v, and the rise time function is assumed to be scale dependent. The partial slip associated to a given wavelength 1/k is assumed to be completed in a time 1/kv, based on simple dynamical considerations. We therefore considered a simple dislocation model (instantaneous slip at the final value), which indeed correctly reproduces this self-similar characteristic of the slip duration at any scale. For a simple rectangular fault with isochrones propagating in the x direction, the resulting far-field displacement spectrum is related to the slip spectrum as ft(w) = F Ca v ky 0 where the factor F includes radiation pattern and distance effect, and Cd is the classical directivity coefficient 1/[1 - v/c cos (0)]. The k-square model for the slip thus leads to the w-square model, with the assumptions above. Independently of the adequacy of these assumptions, which should be tested with dynamic numerical models, such a kinematic model has several important applications. It may indeed be used for generating realistic synthetics at any frequency, including body waves, surface waves, and near-field terms, even for sites close to the fault, which is often of particular importance; it also provides some clues for estimating the weighting factors for the empirical Green's function methods. Finally, the slip spectrum may easily be modified in order to model other powerlaw decay of the radiation spectra, as well as composite earthquakes. Introduction The kinematic rupture models inverted from local or teleseismic seismograms clearly show the complexity of the rupture process. The sources present large heterogeneities in the co-seismic slip (a review may be found in Brune, 1991), and rupture velocity variations are strongly suggested in some cases (e.g., Archuleta, 1982). Unfortunately, these inversions provide pictures of the rupture history on faults with a spatial resolution that is 1216

A Kinematic Self-Similar Rupture Process for Earthquakes 1217 usually not better than about one-third or one-fourth of the typical size of the event, as a result of the usual limited number of the records and to the requirement of some redundancy in the data for a reliable inversion (e.g., Hartzell et al., 1991; Fukuyama and Mikumo, 1993). For example, a 100-km-long fault (magnitude about 7.5) would have presently a maximal resolved wavelength of typically 30 kin, while it would be 300 m for a 1-kmlong fault (magnitude about 4). Therefore, no information on the spectral decay of the slip function can be inferred from these inversions, which do not map the fine details of the slip history on the fault. The complexity of the seismic source body-wave radiation at high wavenumber and frequencies is thus usually characterized by the power law of the decay of its spectral amplitudes at frequencies greater than the corner frequency, after its correction from the attenuation and possible site effects, with no constraint on the phase (except through the requirement of a finite duration of the seismogram). This is, for instance, the classical w-square model (w -2 slope for the far-field displacement) first introduced by Aki (1967), statistically observed for numerous earthquakes (e.g., Houston and Kanamori, 1986), the average w -15 reported by Hartzell and Heaton (1985) for shallow subduction events, or the more recent models including a second corner frequency and intermediate spectral slope w-1 (e.g., Boatwright and Choy, 1992). Several methods have been proposed for synthesizing seismograms, taking advantage of this apparent stochastic nature of the phase at high frequencies. Some methods directly invert in time a spectral function with the appropriate shape and amplitude, with a stochastic phase (e.g., Boore, 1983), or sum adequately a set of empirical Green's functions that includes propagation effects (Wennerberg, 1990). Such methods do not consider the details of the physical origin of the radiated wave from the fault plane, which is considered as a point source. On the contrary, some other methods involve kinematic models of finite faults, with some random rupture process, coupled with numerical or empirical Green's functions methods for propagating the field to the recording site (Papageorgiou and Aki, 1983; Irikura, 1992). A difficulty with the former empirical approach is that it assumes that the source is small with respect to its distance to the station, which is often not true for nearsource records of large earthquakes (distance a few tens of kilometers, magnitude greater than 6.5). Hence, this technique cannot be considered adequate for such conditions, although those are of particular importance for aseismic design, because of the associated high amplitude of the ground motion. The difficulty for the latter approach with kinematic modeling is to find a priori the proper set of random kinematic parameters that will produce synthetic seismograms with a realistic spectrum. For instance, Papageorgiou and Aki (1983) introduce a subevent characteristic size, which leads to a strong characteristic frequency in the radiation. Another example is the pseudo-dynamic model used by Lomnitz- Adler and Lund (1992), which leads to an overestimation of the spectrum, thus requiring a low-pass filtering with an ad hoc rise time. The synthetics generated by the empirical Green's function methods, which assume some distribution of subevents with specific nucleation times and weighting factors, also present the same difficulty in matching the spectral amplitude decay. A different approach was initiated by Hanks (1979), based on the idea that the self-similarity and the spectral law of the seismic radiation should be described by some self-similar rupture process. Hanks assumed that the stressdrop distribution on the fault plane depends on the radial wavenumber k in 1/k ~. In order to obtain a constant rms value of the stress drop, analyzing the scaling law of the radiation, he found n = 2. Following a similar idea, Andrews (1981) used a spectral model for the slip velocity in frequency and wavenumber, assuming a random distribution of this function, which led him to the value of n = 1. Recently, Frankel (1991) proposed to describe the rupture process with a set of cracks showing a selfsimilar distribution in size. This led to a fractal dimension of 2 when the "subevents" fill the area of the "main" event, at each scale level, which corresponds to a stressdrop spectrum in 1/k (n = 1), thus compatible in this particular case with Andrews's results. Although all these authors showed that their model should generate the typical w-square model, the effective calculation of synthetic seismograms and of their spectra was beyond the scope of their studies. Indeed, the timing of the rupture propagation, required for such kinematic modeling, was not analyzed. Following this "self-similar" approach, the aim of the present article is thus to define a kinematic model of rupture presenting a self-similar distribution of rupture parameters (final slip and rupture time) and based on reasonable physical constraints that fits a given spectral law of the far-field displacement (e.g., w-z). This will then shed some light on the physical rupture parameters leading to the observed radiation spectra and provide a method for generating realistic synthetics valid also in the near-source and near-field range, with a straightforward application to the empirical Green's function methods. These aspects will be more extensively developed in subsequent articles. In the present article, we first characterize the spectrum of final slip distribution, and then similarly the stressdrop spectrum, assuming a specific self-similar law. We finally build a kinematic model, assuming a constant rupture velocity and introducing a specific rise time function, and compare the resulting theoretical and numerical radiation spectra to the empirical spectral laws.

1218 A. Herrero and P. Bernard Dislocation Spectrum Versus Fault Length We assume that the spatial spectrum of the co-seismic slip on a hi-dimensional fault of characteristic rupture length Lo, Auto, depends on L0 and k, the radial wavenumber, scaled by the low-frequency level (,)n (1) for k > 1/Lo and a'u~o(k) = a-u~o(o) for k < l/l0. The low-frequency term is relatedto the mean value of the slip on the whole fault ~-ul0, h'ulo(o) = h-ulo Lo z. (2) The dependency of the mean slip value on L0 is a well-known scaling law (Eshelby, 1957; Keilis-Borok, 1959; Knopoff, 1958), and may be expressed as Aum = C--Lo, (3) where Ao- is the mean stress drop, /~ the rigidity of the medium, and C a geometrical factor (C = 1). Now, injecting equations (3) and (2) in (1) gives ~~o(k) = c a----~l~ (+0) n. /.t In order to constrain the value of n, let us consider two activated faults, with lengths L0 and Ll < L0, both following equation (4), and restrict ourselves to wavenumbers k > 1ILl, disregarding wavelengths greater than L1 at this stage. Our main assumption is now that the mean slip amplitudes at the corresponding wavelengths are the same for both events. For example, the spatial fluctuation of the slip observed at a characteristic length scale of 100 m will statistically have a similar amplitude for a 1-km-long rupture as for a 100-km-long rupture. Such an assumption seems reasonable, because these fluctuations are related to the elastic strain threshold of the rock, independent of scale, and typically 10-4 to 10-5. Hence, the formula (4) can also be used for describing the high wavenumber content (k > 1/Ll) of a subarea with size L~ < Lo belonging to the rupture area of the event L0. The number of such subareas is N = (LolL1) 2. Note that these subareas are not equivalent to subevents of similar size obeying the scaling law given by (1), because they do not have the same low-wavenumber amplitude; such subevents would be depleted in seismic (4) moment with respect to these subareas. Considering that the high-frequency slip spectrum of the main rupture L0 is the sum of the high-frequency slip spectra of all the subareas L~ (resulting from the definition of the spatial Fourier transform), this summation may be estimated by considering that the high-frequency (k > 1/Lx) spectral components of the slip distribution on each subarea L~ have an independent phase. Their summation is therefore incoherent, and the resulting spectrum can be estimated by V~ a-ul,(k) = a'ul0(k), (5) where k > l/l1, assuming that the proportion of broken area on a fault segment activated by an earthquake remains statistically the same for all earthquake magnitudes. Replacing both sides of (5) with the formula (4) leads to n -c--l? = c--lg, L 1 b6 /3, which finally gives n = 2, whatever the value of L0 and L1. Injecting this last result in equation (4), we obtain the following general equation of the slip spatial spectrum: Atr L hul(k) = C-- (6) ~k ~" This spectral model for the slip distribution will be referred to as the "k-square" model; its low wavenumber content is controlled by the seismic moment, and its high wavenumber decay exponent results from an assumed and plausible self-similarity of the slip distribution. A numerical example is given in Figure 1, which presents the slip distribution corresponding to the spectrum in (6), assuming a random phase distribution in the wavenumber domain, except for the very first terms at low wavenumber. For simplicity, the phase of the latter has been specified here in order to maximize the slip near the center of the rectangular fault. The high wavenumber characteristics of the slip given by (6) is depicted in Figure 2, which shows a cross section of the slip for three faults with different dimension, each following the k-square model. The slip section at the left corresponds to a fault similar to that in Figure 1. Note that the three faults show a similar amplitude for the corresponding high-pass filtered slip fluctuation (top of Fig. 2). The slip distribution and cross sections of Figures 1 and 2 show fluctuations that look self-similar; at a given wavelength, the ratio of the mean slip to this wavelength is constant, suggesting a strain (or mean stress drop) independent of scale. Other slip distributions have been proposed, as will

A Kinematic Self-Similar Rupture Process for Earthquakes 1219 be seen in the next paragraph. Let us here estimate its spectral slope in one case, which will be used later for the kinematic analysis. The slip spatial distribution is provided by the pseudo-dynamic model of Lomnitz-Adler and Lemus-Diaz (1989), who showed that it could be described by an effective probability of fracture of a site. As each fractured site slips by a given amount, constant over the whole plane and hence independent of the size of the cluster, we conclude that the mean slip on a cluster is independent of the size of the cluster. Moreover, isolating a subarea with a fixed size in a cluster gives a local slip pattern independent of the size of the cluster. These two properties allow us to follow the same line of arguments developed in the present article, and we obtain the corresponding spectrum - L ~ul(k) = Auk Thus, the slip spectrum in Lomnitz-Adler and Lemus-Diaz (1989) is in I/k, thus differing from (6), as sketched in Figure 3. Such a model is not physically suitable, because either the strain (or stress drop) is too low at low wavenumbers, or too large at high wavenum- bers. The adequate stress characteristics will be analyzed in detail in the next paragraph. Stress Drop Spectrum Versus Fault Length We consider here the stress drop on the plane containing the ruptured fault, which should therefore be mostly positive in the ruptured area, and negative (stress increase) outside it. The stress drop spectrum is assumed to depend on the characteristic fault length L and the radial wavenumber, for k > 1/L, as 2(o~L(k) = al~ L P k q, (7) where a is an adimensional constant. The homogeneity of equation (7) implies that p - q = 2, which becomes, for k > 1/L, and for k = l/l, L p A'o'L(k) = a/z (8) ~-~, ~'o (1/L) = a/z L 2, (9).e~ ~ % Figure 1. An example of a slip distribution satisfying the k-square model (equation 6), with the following parameters: C = 1, L = 5.8 km, Ao- = 4 MPa, /z = 3.3 10 ~ Nm. The slip distribution is tapered at the edge of the fault, and the negative values are set to zero. The phase of the lowest wavenumbers is calculated in order to concentrate the slip near the center of the rectangular fault. At higher wavenumbers, the phase is stochastic.

1220 A. Herrero and P. Bernard 0.06 0.06 0.06 0.04 0.04 0.04 0.02 E o.oo v -0.02 0.02 0.00-0.02 0.02 0.00-0.02-0.04-0.04-0.04-0.06 0 1 2 3 4 5 6-0,06 0 1 2 3-0.06 0 1 4 I I I I I P 3 E2 v 1 0 1 2 3 4 5 6 (km) 2.0 1.5 1.0 0.5 0.0 I q I 1 2 3 (km) Figure 2. From left to right, three cross sections of the final slip distributions for three different earthquakes satisfying the k-square model are presented, with the total slip at the bottom, and the corresponding high-pass filtered slip at the top, removing wavelengths larger than 500 m. The earthquake model to the left corresponds to a slip distribution similar to that in Figure 1. The model at the center corresponds to a source size two times smaller, and the model to the fight to a size four times smaller, corresponding, respectively, to a seismic moment 8 and 64 times smaller. Note the self-similarity of the slip distribution, and the values of the filtered slipamplitudes, independent of the size of the generic earthquake. They are a few centimeters, for a 500-m wavelength, corresponding to a strain between 10-4 and 10-5. 1.0 0.5 0.0 0 1 (km) LogeS) ~ 3 ~ 2 ~ / k Log~) deduced from deduced from Hanks (1979) Frankel (1991), Andrews (1981), deduced from Lomnitz-Adler & I.~muz-Diaz (1989) and the "k-square" model of this study Figure 3. This picture summarizes the different results for the slip distribution spectrum that we infer from the results of Hanks (1979), Andrews (1981), and Lomnitz-Adler and Lemus-Diaz (1989), and a particular solution of Frankel (1991), which is consistent with the k-square model introduced in the present article. We consider two distributions on faults with characteristic length L0 (solid line) and La = Lo/2 (dash line). N is the ratio of the surfaces (N = L~/L~).

A Kinematic Self-Similar Rupture Process for Earthquakes 1221 which does not depend on p. The mean stress drop on the whole plane containing the fault is zero (the stress has simply been transferred from the ruptured area to the surrounding), and the mean stress drop on the ruptured area of the fault is approximated by Ao- L = 1 ~rl(1/l), A~r L = a/z. (10) In the following, the stress drop will refer to the function restricted to the rupture area, if no other precision is given, and hence will be zero on the fault plane outside the rupture area. From (10), the mean stress drop is therefore, as expected, independent of L. Before estimating the value of the parameter p, let one first estimate the rms value of the stress drop, in order to recall the main results by Hanks (1979) and Andrews (1981). This function is given by 2 _lf(a~2(x,y)dxdy ' (11) AO'rms'L = S where S is the fault area, which can be rewritten, using Parceval's theorem, as 2 1 fo += AOvrms, L = - S ~'o-2(k) kdk. Using (8) and (9), (12) becomes AO-rms'L 2 = S [fo Thus, ifp < 1, Jr- a2tx2l4kdk (12) az/a, 2 L2P /L k-~- 2p dk " (13) mo'rms,l a2]/,2 [1+ ] 1,,4, 1 p The choice p = 0 in (8) corresponds to the following solution given by Hanks (1979): The corresponding rms value is ~-o (k) = a ~. (15) Ao~s,L = a/,, (16) which is constant, as was required by Hanks. Let us point out from (14), however, that this last property, which justified Hanks's choice of the value of p, is also verified for any value p < 1. Hanks's solution is therefore a priori not more appropriate than any solution with 0 < p < 1. In Andrews (198_1), the two-dimensional stress-drop power spectrum is Ao -2 ~ l/k 2, giving, in the present notation, p = 1, which implies a divergence of (13). ff p = 1, equation (8) gives L ~o (k) = a/z-. (17) k One can study the rate of the divergence of (13) in this case, by truncating the integral in equation (13) at k = km~. This upper value kmo~ could be connected with the size of crystal grains in rocks, or even to atomic size, where the validity of the scaling law is likely to break down. Equation (13) becomes AO'rms,r + log (kmaxl). (18) Extreme values of the rms stress drop are found by maximizing the product kma x L, taking a large fault dimension L = 105 m and a large wavenumber km~ = 102 m -~ (atomic size). The rms stress drop is then about only six times larger than the reference value provided by (16) for this length range of 15 orders of magnitude. Hence, this divergence is extremely slow, and physically acceptable in the whole seismic range; the value p = 1 may be a priori as acceptable as the values 0 < p < 1. Let us now calculate p by following the same arguments for the stress distribution on a fault as we did above for the slip distribution. One considers two faults with length L0 and L1 < L0, corresponding to two characteristic seismic ruptures obeying (8). A subarea of the fault L0, with a dimension L1, is assumed to have the same stress fluctuation, for wavenumbers larger than 1/Lt, as those for the characteristic fault L~. The incoherent summation of the high wavenumber stress drop of all the subareas with dimension L~ in the main fault should equal its high wavenumber spectrum. Hence, for a wavenumber k > 1/L~, (19) where N is the number of L l dimension faults needed to reconstruct an L0 dimension fault, i.e., N = Lo/LI.2 Using equation (8) in both sides of equation (19) gives Lo L~ G L--~ a/z k2---- ~ = a/~ kz_p,

1222 A. Herrero and P. Bernard which provides p = 1. Hence, with the assumption that the rupture characteristic at a given scale smaller than the fault dimension does not depend on the size of this fault, the stress-drop spectra is given by equation (17), which agrees with Andrews's (1981) result. This result is also equivalent to the particular solution given by Frankel (1991) when his crack distribution has a fractal dimension D = 2. Combining (6), (10), and (17) provides the following relation between stress drop and slip spectra, for k >l/l, ~crl(k) = k ~ ~ul(k). (20) C Note that this relation is scale invariant, as L does not appear in it. In fact, this relation is an approximation of the exact relation given for any k by Andrews (1980) [see equations (23) and (25) in his article], which takes into account the phase difference between the stress and slip spatial Fourier transforms, and the slight difference between mode I/and mode III coupling coefficients between stress and dislocation on the fault plane. For the present study of scaling law, the relation (20) may be considered as a reasonable approximation, which shows the equivalence of the two assumptions which lead independently to (6) and (17). Following Andrews (1980), the validity of the approximation in equation (20) may be extended for the low wavenumbers k < 1/L, considering now the stressdrop function defined on the whole plane, and not restricted to the ruptured area as above. Hence, the slip spectrum being flat for k < 1/L, equation (20) leads to a stress-drop spectrum on the whole fault proportional to k for k < 1/L. For k = 0, we thus get a zero stress global stress drop on the plane, which is a well-known property. Using (20), the solutions for the stress spectral decay given by Hanks (1979), Andrews (1981), and a specific solution of Frankel (1991) can be rewritten in terms of slip spectra and are summarized in Figure 3. A Kinematic Model Based on the k-square Slip Model In the preceding paragraphs, only the difference between the initial and final slip and stress state of the ruptured fault was considered, and nothing was stated about the rupture process itself. Let us now introduce a kinematic of the rupture, in order to calculate, in the farfield approximation, the radiated spectrum associated to the final slip distribution depicted by (6). The first fundamental parameter that has to be specified is the rise time distribution on the fault plane. It is defined as the time ~- during which the fault locally slips. For most studies assuming a propagating dislocation, the rise time is distributed homogeneously in space, and associated with a constant slip velocity; its role is to smooth out the unrealistic effects of a slip discontinuity at the rupture front, by low-pass filtering the radiation at the frequency 1/z with an to -1 filter. This approximation is convenient for looking at frequencies up to 1/'r, but is not aimed to model the higher frequencies. At a smaller wavelength, the slip velocity may indeed show significant spatial and temporal variations during the rise time. When dealing with a continuous distribution of slip wavelengths higher than 1 Iv,r, this classical concept of rise time with constant slip velocity is thus not suitable anymore. Let us consider a final slip distribution on a fault (length L0) given by (6), and extract the slip contribution around a given wavenumber k = 1/L greater than 1/L o. The main fault may then be divided in a set of subareas with characteristic lengths L, associated with some characteristic narrow band-filtered slip proportional to the bandwidth and to the slip spectral amplitude at this wavenumber. From simple dynamic consideration deduced from crack rupture modeling, the duration of such a partial slip on a given subarea S should be of the order of L/v, where v is the rupture velocity and L the size of S. Indeed, when the rupture front passes through this subarea, the slip starts increasing, but as soon as the rupture reaches its border (after a time L/v), a "partial" stopping phase is expected to travel back in S, due to the local and temporary slip velocity decrease at this barrier. This stopping phase will interact with the higherstrength periphery of S, and the resulting wave, which has a characteristic frequency of the order of v/l, will heal the slip motion at this frequency. This speculative dynamic process is suggested by the dynamic studies of seismic rupture (e.g., Madariaga, 1977). The long wavelength--or low frequency--part of the slip on S will not be affected by the diffraction effects at the edge of the subarea S, and will therefore continue to increase, until the complete locking of S [at the end of the total rupture, for classical crack models, or after a shorter time, for self-healing rupture models (Heaton, 1990)]. This is equivalent, in the discrete case of Frankel's 1991 model, to the assumption of an inverse proportionality relationship between the subevent dimension and the comer frequency of the subevent radiation, whatever the scale of this subevent. In other words, we suggest that the rise time is not a simple function of space; it has to be considered in the wavenumber-time domain. Paradoxically, a first-order approximation of the effect of a such a nonclassical rise time function is provided by a propagating rupture front at which the slip instantaneously reaches its final value, i.e., a very classical dislocation model, associated with a heterogeneous final slip distribution. Indeed, in such a kinematic model, the subarea S is crossed by the rupture front in the time L/v, and thus the associated partial slip at this wavelength is completed in

A Kinematic Self-Similar Rupture Process for Earthquakes 1223 exactly this time, which is precisely what was required by the qualitative dynamical considerations above. Of course, the rupture development in the subarea L seen at frequencies close to v/l is certainly more complicated than what is described here by the sweeping of the rupture front line, but these complexities are expected to be second-order effects when looking at the spectral amplitude at this frequency. The other fundamental parameter that has to be specified is the rupture velocity. Most models used for direct modeling or inversion use constant rupture velocity. This constraining assumption needs to be discussed. Indeed, this strong hypothesis seems to be contradicted by some inversions of seismograms (e.g., Archuleta, 1982). Furthermore, it is well known that the body-wave radiation is sensitive to both the final slip heterogeneity and the rupture velocity fluctuation (e.g., Bernard and Madariaga, 1984; Spudich and Frazier, 1984). Thus, one may argue that the rupture velocity fluctuation has a significant effect on the high-frequency radiation. In the extreme case of a smooth final slip distribution (for instance, a bell-type function on the fault plane) and a fluctuating rupture front velocity, the high-frequency spectra would be generated only by these fluctuations. However, it may be suggested that a smooth final slip requires some smooth dynamic development, and hence is expected to produce a low level of high-frequency radiation. On the other hand, a very heterogeneous slip velocity field is likely to be accompanied by both heterogeneous final slip and irregular rupture front propagation (e.g., Cochard and Madariaga, 1994). The knowledge of the relative contribution of these two coupled processes to the radiation is of importance for understanding the dynamics of faulting, but is, however, beyond the scope of our study. For the purpose of the present study, as both processes have a similar effect on the high-frequency radiation that we aim to simulate, and because from the numerical point of view heterogeneous slip distribution with constant rupture velocity is easier to handle than the variable rupture velocity models, we use in the following the constant rupture velocity assumption. The physical consequences of this assumption will be reevaluated in the discussion. In the following, the kinematic model is then defined by a rupture front propagating at a constant rupture velocity, associated with a k-square final slip function reaching its final value instantaneously. The far-field displacement u at a station is expressed by the following integral over the fault surface S of the slip velocity Au: on the fault, on the elastic properties of the medium, and on the radiation pattern (see, e.g., Aki and Richards, 1980), and tp is the wave travel time. In order to simplify the analytical solution of the problem, the station is assumed to be in the far-source range (distance greater than the source dimension), the rupture front is a linear segment parallel to the z axis (Haskell model) propagating in the x direction, and the rays connecting the source to the station are contained in the (x, y) plane, making an angle 0 with the x axis, as depicted in Figure 4. The isochrone concept, introduced by Bernard and Madariaga (1984) and Spudich and Frazier (1984), allows us to rewrite equation (21) as an integral over the isochrone u(t + r) = F f0 l dz v CdAu(vtCa, z). (22) In this simple geometry, Ca is 1/(1 - v/c cos 0), where c is the wave velocity, and the isochrones coincide with the successive rupture front positions. This leads, in the frequency domain, to a(w) = F Jo dz v Ca Au(vtCd, z) e-i~tdt. (23) Changing the variables in the integral, with kx = oj/ (vcd) and k~ = 0, we find the spectrum of the far-field displacement a(o,)=f kx=ca (24) This expression was obtained by Bernard (1987). A power-law decay (in wavenumber) for the slip spectrum thus simply produces the same power-law decay (in fre- X station direction u(t) = fs dxdzfau(x, z, t - tp), (21) where F is a fa.ctor depending on the distance to the point Figure 4. Geometry of the simplified kinematic model used for the analytical calculation. See text for details.

1224 A. Herrero and P. Bernard quency) for the body-wave radiation. Note also the compression of the spatial spectrum in the direction orthogonal to the isochrone, due to the directivity. Thus, assuming now a k-square model for the slip (equation 6), the high-frequency displacement spectrum (24) becomes Acr 2 2 1 a(w) = FC ~ Lv Ca -~22, (25) which is the "w-square" model. This result is in agreement with the spectral predictions from continuous and discrete slip distribution models (respectively, Andrews, 1981 and Frankel, 1991, in the case where the sum of the subevents areas equals the mainshock area). A more general geometry of the isochrones will not change qualitatively this last result, which is the w -2 decay of the displacement spectrum. The C 2 factor associated with the w-square model was already pointed out by Joyner (1991) for the simpler case of a one-dimensional kinematic source. The following numerical example considers the k-square slip distribution on a rectangular fault, depicted in Figure 1, with a rupture front expanding circularly from the hypocenter, located at the edge of the fault. We compute the far-field S acceleration in an elastic homogeneous half-space, using the isochrone concept associated with the kinematic description used by Yoshida (1986). One difference between our approach and Yoshida's is that we work in the near-source range, involving, for a finite source, variable distances and ray orientations, whereas Yoshida works in the teleseimic range. A typical result is presented in Figure 5, for a magnitude 6 earthquake recorded at 13 km. The far-field S acceleration generated by the kinematic model in a homogeneous half-space shows the typical flat spectrum above the comer frequency, as described by the w-square model. Moreover, the calculated amplitude of the spectrum and of the peak acceleration are typically those expected for this magnitude and distance. In the case of the slip spectral distribution proposed by Lomnitz-Adler and Lemus-Diaz (1989), which de- 0.6 10 0 4 0.3 0.0-0.3 E 10-1 10-2 -0.6 10-3 2 3 4 5 6 7 8 9 10.2 101 100 101 sec f 10 2 (a) ~ nucleation point 10 Km station (b) 10 Km Figure 5. (a) S-wave acceleration and its spectrum in a homogeneous elastic half-space due to the radiation of a kinematic model described by the slip distribution of Figure 1 and a constant rupture velocity. The station is 13 km away from the source, as sketched in (b). (b) Map of the source (rectangle) and station (dot) of (a). The fault plane is dipping 55 toward north (top of the figure). The star is for the nucleation point.

A Kinematic Self-Similar Rupture Process for Earthquakes 1225 cays as 1/k as demonstrated above, a kinematic model with a dislocation propagating at a constant rupture velocity would provide, with equation (24), a 1/to radiation of the far-field displacement spectrum. This is precisely what is obtained by Lomnitz-Adler and Lund (1992), who used this particular slip distribution, as shown by the to slope of the acceleration spectrum in their Figure 9a. This result led these authors to low-pass filter the time series with an ad hoc, scale-independent rise time function with constant rupture velocity in order to reproduce the co-square model. Discussion Let us further discuss our assumptions of self-similar rise time and constant rupture velocity. Let us consider first a more classical rise time function, with a constant rupture velocity lasting a time r, and keep the k-square model for the slip. Its effect is to filter the bodywave spectrum with an co-1 filter for frequencies higher than 1/r. A constant rupture velocity would then produce an co-3 decay of the body waves at frequencies above 1/r, which is usually not observed. As the k-square model produces the maximum acceptable slip fluctuations (i.e., the spectral slope cannot be smaller than 2), because otherwise the stress drop would increase dramatically at short wavelength, the only reasonable way to reach the highfrequency level of the co-square model would then be by introducing very strong rupture velocity fluctuations in the kinematic model, which would be the dominant source of high-frequency radiation. However, it would be unrealistic to assume strong fluctuations of the rupture front at all scales and to keep at the same time a constant slip velocity behind this rupture front, because the local and temporary accelerations of the latter are expected to strongly interact with the slip velocity. Behind the rupture front, slip velocity fluctuations are thus expected. Their spectrum in the frequency-wavenumber domain should be in 1/k 2 and 1/092, according to Andrews (1981), for generating an co-square model. It is straightforward to show that the slip velocity of our k-square kinematic model obeys this property. Constructing more sophisticated kinematic models with such a frequency-wavenumber spectrum, but with a nonzero slip velocity confined to some slipping band sweeping the fault plane, such as the self-healing pulses proposed by Heaton (1990), would certainly be more realistic than our model, but would not change our basic result on the radiation spectrum. The main improvement that such models would bring concerns the directivity effects, as shown in detail by Bernard and Herrero (1993). Furthermore, for wavelengths larger than the width of the slipping band, the latter can be considered as a line, and the dislocation model with instantaneous slip and self-similar rise time proposed in the present article remains a good approximation. Independently, future refined seismogram inversions or field studies near ruptured faults might allow the construction of an empirical standard spectral model for the slip distribution, which might well show a 1/k m power law spectral decay at high wavenumber. The results of the present article then predict that m should be larger than or equal to 2. Moreover, if m were to be larger than 2, the contribution of the rupture velocity fluctuations would largely dominate the effect of slip heterogeneities to the radiated spectra in order to reach the co-square radiation spectra, because the contribution of the slip heterogeneity would be in co-m, much smaller than the observed spectra. Alternatively, if m were equal to 2, thus validating the k-square model, then the co-square model would be fitted without the need of any velocity fluctuations, as shown by (25), and hence the true velocity fluctuations would have an effect on the radiation smaller or at most equal to the effect of the slip heterogeneities. Whatever the real physical process dominating the high-frequency radiation, as far as generating realistic synthetic seismograms, the kinematic model with a constant rupture front velocity is adequate, because the k-square slip distribution may simply be considered as an ad hoc function, including the effects of both real slip and slip velocity distribution, and those of local rupture velocity fluctuation. Once the slip distribution is fitted to the required spatial spectrum, the kinematic model is adequate for use with any Green's function, in the frequency range from zero to the cutoff frequency related to the fault grid size. Thus, not only body waves, but also surface waves and near-field terms could be calculated from this model. The complete field may thus be calculated, in particular for sites that may be very close to the fault, a case relevant for engineering applications for which the existing stochastic kinematic models are usually not suitable. A few studies show that some earthquakes have a spectral radiation significantly different from the co-square model (e.g., Hartzell and Heaton, 1985). We may then propose more sophisticated slip spectra, presenting, for instance, a k -~ slope between two length scales, which will simply result in an co-' slope of the displacement between the two associated corner frequencies. This may be required for modeling composite earthquakes (Boatwright, 1988) or for presenting some dominant wavelength (characteristic asperity or barrier interval), as in Papageorgiou and Aki (1983), and will result in a characteristic frequency estimated by equation (24). A numerical example is presented in Figure 6a, which shows a slip distribution with two main patches separated by a barrier. The slip differs from that of Figure 1 only in the phase characteristic of the first low wavenumbers. This slip model has been used to compute the S-wave acceleration presented in Figure 6b, as in Figure 1. The ra-

1226 A. Herrero and P. Bernard diated body wave and the peak amplitude are similar to those of Figure 5. Another important consequence of the spectral laws depicted in this article concerns the empirical Green's function methods, which are more and more widely used in the engineering community for generating realistic synthetic seismograms. Indeed, one may consider that our kinematic model uses a "numerical Green's function" at the scale length of the grid, scaled by a "weighting factor" which is the slip value at this point. In fact, this analogy is the theoretical basis of the method. Constructing a realistic radiation spectrum thus requires a very specific distribution of weighting factors of the Green's functions (whether empirical or numerical), which is shown by (6) if the rupture velocity is held constant. These factors should follow the k-square model (or a similar model, as explained above) for modeling frequencies lower than the corner frequency of the Green's function. Finally, it should be recalled that the various spectral laws analyzed here attempt to describe only the statistical mean behavior of stress and slip distribution of a large set of seismic ruptures. For a single event, the spectrum may show a significant departure from these laws. Moreover, the phase of the spatial slip distribution may be far from stochastic, showing, for the same rup- el % (a) 0.6 0.3 10 0 10"1 I,,... H;... ;... '~ 0.0-0.3 I= 10 `2-0,6 I I i 2 3 4 5 6 7 8 9 s e c (b) 10.3 10-2 10-1 10o 101 102 f Figure 6. (a) Same as Figure 1, except for the phases of the lowest wavenumbers. They form two patches with a barrier (dislocation of value zero) at the center of the fault (hidden behind the left patch). (b) Same as Figure 5, for a slip described in (a). The waveform of the signal is quite similar to that in Figure 5, except at low frequency.

A Kinematic Self-Sirnilar Rupture Process for Earthquakes 1227 ture, areas of high stress fluctuation, and areas of low stress fluctuation (smooth and rough faulting areas). The extreme case would be that the co-seismic slip is smooth in most of the fault area, except along some lines where strong spatial gradients exist perpendicularly to these lines, which would correspond to geometrical discontinuities of the fault surface (e.g., steps, jogs, etc.). These bartiers would then account for most of the high-frequency radiation, as shown by Bernard and Madariaga (1984). This is expected to significantly affect the seismogram, which may show some departure from stationarity, in particular in the near-source distance range where the site is strongly sensitive to a restricted area of the fault. The k-square model with a stochastic phase may then be improved, considering fluctuations around the k -2 spectral slope and constructive phase correlation. Such a model would then help to analyze the nonstationarity of the highfrequency waveforms in terms of the spatial variations of the strength distribution on the fault, which may be related to the mean stress drop in (6), defining the absolute level of the slip spectrum. It may thus help to develop some inversion method of records at frequencies much larger than the comer frequency. In conclusion, the hypothesis of a self-similar distribution of the slip led us to define a k-square model (k -2 spectral decay) for the co-seismic slip distribution in the fault plane, which corresponds, for the stress drop, to a k -1 spectral decay, following Andrews (1981). We showed that the k-square model, combined with the assumption of a constant rupture velocity and of a scaledependent rise time, results in a kinematic model radiating the classical w-square model. This kinematic model of seismic rapture may thus be suitable for generating realistic synthetic seismograms and spectra at any frequency and any distance from the fault, if the appropriate earth structure and Green's function is introduced. This model may easily be adapted for modeling any radiation spectra differing from the w-square model, by using the equality between the spectral slopes of the slip and of the body-wave displacement. Acknowledgments We would like to thank R. Madariaga for his careful reading of the original manuscript, and two anonymous reviewers, for their important comments and advice. We also thank A. Turmakin and D. J. Andrews for their discussion on the article during Erice Summer School (Italy). This work has been supported by INSU (Contract Number 500978), CEC (Epoch Contract Number CT910042), and BRGM (Direction de la Recherche). IPGP publication 1310. References Andrews, D. J. (1980). 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