Finite Fault Modeling of Strong Ground Motions Using a ... - CiteSeerX

Bulletin of the Seismological Society of America, Vol. 95, No. 1, pp. 225–240, February 2005, doi: 10.1785/0120030163

Finite Fault Modeling of Strong Ground Motions Using a Hybrid Deterministic–Stochastic Approach by Francesca Pacor, Giovanna Cultrera, Andres Mendez, and Massimo Cocco

Abstract

A hybrid deterministic-stochastic method (DSM) is developed to calculate synthetic time series of ground accelerations radiated from an extended source. The main goal of the proposed methodology is to include in the classical pointsource stochastic method (PSSM) the effects of the rupture propagation on a finite fault. This purpose is achieved through two important modifications of the PSSM technique. First, the envelope does not have a predetermined functional form; rather, it is calculated deterministically following the isochron formulation with a kinematic rupture model. Second, we have generalized the various parameters of the pointsource ground motion spectrum to account for the extended fault: corner frequency, distance from the fault, and radiation pattern are evaluated through the kinematic modeling. The guiding principal in all these modifications has been to develop a robust methodology capable of capturing the complexity of near-source ground motion even when input information about earthquake source, propagation medium, and site characteristics are of a very schematic nature. We show that the synthetic envelope contains the required information on the rupture process on extended fault, such as directivity effects and azimuthal variations depending on the source-to-receiver geometry. The method’s capability is demonstrated by modeling strong ground motions of the 1992 Mw 7.3 Landers, California, earthquake and comparing them with the recorded accelerograms, which are clearly affected by directivity effects. The proposed technique reproduces the main characteristics of strong-motion recordings, and can be implemented using only a limited number of parameters to describe the source (dimension and geometry), the propagation medium (wave velocities and layers), and the site effects (transfer function). These characteristics are important for a methodology aimed to simulate groundshaking scenarios for which a more complete description of the faulting process is not available.

Introduction 2003 and references therein), as well as multi-pathing and scattering that occur during the propagation in heterogeneous media. Important progress has been achieved recently in the deterministic modeling of the rupture process and seismic wave propagation. These methods allow the simulation of radiated ground motion using either kinematic (Olson et al., 1984; Bernard and Madariaga, 1984; Spudich and Frazer, 1984; Spudich and Archuleta, 1987; Mai and Beroza, 2002) or dynamic source models (Das and Boatwright, 1985; Inoue and Miyatake, 1998; Peyrat et al., 2001; Guatteri et al., 2003). In the kinematic simulations the spatio-temporal distribution of slip or slip velocity is assigned on the fault plane. Although these methods are important in improving our understanding of the earthquake rupture process, their applicability to predict strong ground motions for engineering

The use of synthetic time series for evaluating groundshaking scenarios and for engineering applications requires the modeling of strong ground motions for a wide range of frequencies, seismic moments, and distances. This goal is difficult to achieve for near-source simulations of moderateor large-magnitude earthquakes because the prediction of realistic time series requires the capability of describing the extended source, together with the wave propagation from the source to the observer, over the entire frequency band. Moreover, the recorded data are incoherent for most frequencies of engineering concern (Hanks and McGuire, 1981; Boore, 1983; Bernard and Zollo, 1989; Cocco and Boatwright, 1993). The incoherence is the result of the heterogeneity of the rupture propagation process (Madariaga, 1976, 1983) and the stress release (Madariaga, 1977; Boatwright, 1982; Boatwright and Quin, 1986; Guatteri et al., 225

226 applications is limited: the high-frequency radiation is controlled by the small-scale heterogeneity of the rupture process, which cannot be prescribed a priori and requires a stochastic description. A further limitation is that poor knowledge of the wave propagation path does not allow the theoretical computation of reliable high-frequency Green’s functions. For all these reasons, different approaches have been developed to simulate broadband strong ground motions for the evaluation of shaking and damage scenarios. Some of them use recorded (Hartzell, 1978; Irikura, 1986; Hutchings, 1994) or synthetic (Kamae et al., 1998; Pitarka et al., 2000 and references therein) time series of small earthquakes as Green’s functions, and adopt proper schemes for scaling and adding them to represent the rupture propagation on the fault. Other methods combine theoretical Green’s functions with a wide range of models to represent the seismic source. For example, Zeng et al. (1994) and Zeng and Anderson (1996) represent the source process in terms of a random distribution of overlapping subevents whose dimensions are determined by a fractal distribution on the fault plane. When the rupture front passes, these subevents are triggered and radiate a displacement pulse of the type predicted by a rupture on a circular crack. Other methods describe the source through a deterministic model, modeling the incoherent contribution to the simulated seismic radiation with heterogeneous or random distribution of particular source parameters. For example, Herrero and Bernard (1994) and Bernard et al. (1996) proposed a kinematic rupture model based on a self-similar distribution of coseismic slip. The corresponding amplitude slip spectrum decays with a prescribed power law (kⳮ2) in wavenumber, while the phases are assumed to be random. The model generates realistic broadband accelerations characterized by a x-square spectrum. When a finite rise time is considered, frequency-dependent directivity effects are also simulated. Finally, the so-called stochastic methods use a partially stochastic description of both source and path. These methods have been proposed in past years in order to simulate the high-frequency seismic radiation using a seismological model to constrain the main feature of an earthquake. For instance, the point-source stochastic method proposed by Boore (1983, 2003a; hereinafter PSSM) consists in a windowed random time series whose spectrum matches, only on average, a specified Fourier amplitude spectrum based on seismological model of source, path, and site effects. White noise is windowed by an envelope function represented by a simple analytical expression. Then, the spectrum of the normalized transient time series is multiplied by the specified ground motion spectrum and back-transformed to the time domain. In this article we propose a method which extends the PSSM by including the effects of rupture propagation over a finite fault, such as the rupture directivity. Several studies showed that directivity controls the azimuthal variations of ground-motion amplitudes, duration, and frequency content

F. Pacor, G. Cultrera, A. Mendez, and M. Cocco

at distances within several fault lengths (Boatwright and Seekins, 1997; Somerville et al., 1997). Our goal is to reproduce the main features of recorded accelerograms (signal duration and frequency content, for instance) in this range of distances with sufficient accuracy to be useful in engineering design. In our method, the envelope function is computed by means of a kinematic source model following the isochron formulation (Bernard and Madariaga, 1984; Spudich and Frazer, 1984) instead of a simple analytical expression. The amplitudes of the simulated ground accelerations are controlled by a specified point-source-like amplitude spectrum (such as a x2 spectrum) whose parameters are evaluated through the kinematic model. We refer to this method as the “deterministic-stochastic method” (DSM), because it joints the stochastic and the deterministic approaches in predicting ground accelerations. Recently several stochastic methods have been extended to overcome the limitation of the point-source approximation in the widely applied PSSM technique. Most of methods for simulating high-frequency motion from extended ruptures consider the source as a superimposition of small events, to be defined in time and space. In the method of Beresnev and Atkinson (1997, 1998, 2002), for instance, the fault plane is discretized into a few subfaults behaving as an omega-square point source and whose radiation is computed using the PSSM method. The contributions from all elements are lagged and summed at the receivers to reproduce the strong-motion record of an earthquake of specified seismic moment. Conversely, the DSM method does not treat the extended fault as a sum of point sources, and bypasses the summation of subevent radiation using the isochron formulation. The DSM method follows the same philosophy of the PPSM method: it is simple and fast from the computation point of view, and it is useful for simulating the frequencies of engineering interests allowing the realization of a large number of plausible rupture scenarios. The technique can reproduce the mean features of near-source ground motions, as the acceleration envelopes and spectral content. However, the corresponding time series should be considered representative of these mean characteristics, as the phase is not physically constrained. In this paper we explain in detail the proposed methodology describing the physical meaning of the envelope, its deterministic computation, and the modification of the source spectrum to take into account the effects of the finite fault. Finally, we show an application of the DSM method to simulate near-source recordings of the 1992 Mw 7.3 Landers earthquake strongly controlled by directivity effects.

Methodology The main goal of the proposed approach (DSM) is to extend the classic PSSM method to simulate strong ground motion by including the effects of rupture propagation over a finite fault. This methodology was originally proposed by

Finite Fault Modeling of Strong Ground Motions Using a Hybrid Deterministic–Stochastic Approach

Mendez and Pacor (1994) and Cultrera et al. (1995), who first suggested a modification of PSSM method. In extremely schematic form, the synthesis of any time series is a four-step procedure (Fig. 1): 1. an acceleration envelope radiated from an extended fault is generated through isochron theory (Spudich and Frazer, 1984) with a simple kinematic rupture process; 2. a time series of Gaussian white noise is windowed with the deterministic envelope, which is smoothed and normalized so that the integral of the squared envelope is unity; 3. the windowed-noise time series is transformed into the frequency domain and multiplied with a point-source-like amplitude spectrum. The parameters of reference spectrum (i.e., corner frequency, distance from the fault, and radiation pattern) are evaluated through the kinematic model to capture the finite-fault effects; 4. transformation back to the time domain. In a typical application of this modeling technique, large number of synthetic time series are generated such that, on average, the spectral properties of the time series mimic those of earthquake ground motion. Our approach follows the PSSM procedure except for two important modifications (Fig. 1). First, the envelope does not have a predetermined functional form; rather, it is calculated deterministically based on a plausible rupture model on extended fault (step 1). Second, we have generalized the various parameters of the point-source ground motion spectrum by modeling them to account for the extended

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fault model (step 3). The guiding principle in all these modifications has been to develop a robust methodology capable of capturing the complexity of near-source ground motion, even when input data regarding earthquake source, propagation medium, and site characteristics are of a very schematic nature. The remainder of this section is divided into three parts. We first review the concept and previous studies of groundmotion envelope. The goal is to provide a rationale for its use as a stable descriptor, in aggregate, of the small-scale processes occurring during earthquake rupture and subsequent wave propagation. The details of the calculation of the ground-motion envelope are dealt with next. We conclude with a description of our proposed generalization of a pointsource spectrum to a finite-source spectrum. Ground Motion Envelope In our method, the main features of ground-motion time series associated with the rupture propagation on extended faults are modeled through the envelope time function. The envelope E(t) of a time series s(t) is defined as the absolute value of the complex envelope function: E(t) ⳱ |s(t) ⳮ iH[s(t)]| where H[s(t)] is the Hilbert transform of the function s(t). The properties of the Hilbert transform yield an easy interpretation of the complex envelope: its absolute value E(t), or instantaneous amplitude, is always positive and tangent to the peaks of |s(t)|, and it agrees with amplitudes measured

Figure 1. Scheme of DSM method: white noise windowed with the deterministic envelope (steps 1 and 2); FFT multiplied with a point-source-like amplitude spectrum (step 3); IFFT (step 4).

228 at signal peaks (Farnbach, 1975); its phase is related to the instantaneous frequency, calculated from zero crossings of the signal s(t), through the time derivative of the complex envelope phase. The separation of amplitude information from phase information contained in the original time series was first used by Farnbach (1975) to study radiated seismic waveforms. Afterwards, several authors proposed the use of the acceleration envelope for ground-motion modeling (Midorikawa and Kobayashi, 1978; Midorikawa, 1993; Cocco and Boatwright, 1993; Boore, 2003b). In the hypothesis of incoherence of high-frequency ground acceleration, the square of the time series, or equivalently, the squared envelope, equals the sum of the squared acceleration pulses radiated by different segments of the rupture front, which arrive at the receiver at the same time. It follows that the ground acceleration amplitudes produced by an extended fault can be synthesized by summing the envelope functions coming from each subfault. These studies pointed out that acceleration envelope time series are easier to model than the acceleration time series, either using recordings of small earthquakes (Midorikawa and Kobayashi, 1978), time functions from empirical relations (Midorikawa, 1993; Joshi et al., 2001), or analytical models (Cocco and Boatwright, 1993). In particular, the study of Cocco and Boatwright (1993) demonstrated that the acceleration envelope is related to the rupture process on extended faults as perceived by the receiver, and can be computed by a simple expression for the instantaneous power, including azimuthal effects due to the source-receiver geometry (radiation pattern and directivity). An important statement of their model is that the acceleration envelope is proportional to the instantaneous power of ground acceleration. To emphasize the relation between energy and envelope, we compare in Figure 2 the time distribution of energy of a real accelerogram and its envelope recorded during the 1980 Mw 6.9 Irpinia earthquake (southern Italy). We filtered the time series using progressively lower values for the highfrequency cutoff, and we computed the cumulative energy of the waveform and the envelope of the filtered seismogram. The dependence of the normalized Arias intensity (Husid plot; Husid, 1969) on the frequency content of the accelerogram mainly appears in the short time-scale roughness of the curve. The shape of each curve is similar, although the final total energy content varies considerably. Similarly, the high-frequency oscillations of the envelope are riding on a lower frequency signal, and significant energy release remains similar both in duration and shape. In other words, both Husid plots and envelopes exhibit a component that is resilient, or robust, to the high-frequency content of the accelerogram. To the extent that the envelope, or lowerfrequency component, represents a spatial aggregate of small-scale source and propagation processes, then a methodology for its synthesis would be opportune for extending the PSSM technique to finite faults. The theoretical basis for such a methodology, as discussed next, is the isochron for-


mulation of Bernard and Madariaga (1984) and Spudich and Frazer (1984). Deterministic Envelope Function DSM computes the synthetic envelope, solving a simplified formulation of the representation theorem through the isochron theory in the following way:

1. identification of the locus of points on the fault for which the emission of seismic radiation is characterized by the same travel time to the site of interest (isochrons); 2. computation of simplified Green’s functions G for each point on the fault; 3. computation of the envelope E at a given site: the Green’s function of each point of the fault is multiplyed for a slip distribution U, and these products are summed in the order predicated by the isochrons. The first step in the generation of synthetic envelopes is the identification of the locus of points on the fault for which the emission of seismic radiation is characterized by the same travel time to the site of interest. These loci are called isochrons; they depend on both the rupture time of each point of the fault (unique for all sites) and the time that the radiation takes to travel from the ruptured point to the site (a time which varies from site to site, and is dependent on the propagation medium). A rupture scenario is simply described by specifying a nucleation point on a rectangular fault plane, from which the rupture propagates radially outward with a prescribed rupture velocity. It is possible to perturb the rupture velocity from its mean value adding a small stochastic component. This concept is illustrated in Figure 3, for the case of a strike-slip vertical fault embedded in a homogeneous medium. The rupture scenario is a circular rupture nucleating at the southern edge of the fault and propagating northwards with a constant velocity (Fig. 3a). The main point of this illustration is that the isochron formulation naturally takes into account directivity effects (Fig. 3b). The rupture is perceived by an observer at ST10 to be accelerating towards the site. The corresponding isochrons show an aggregate of radiation from the rupture of a large portion of the fault arriving in a short time period; therefore, one would expect a strong arrival in the initial ST10 acceleration record. The observer in the opposite direction (ST190) will perceive the rupture front moving away and causing longer duration but lower amplification of the correspondent ground acceleration. In contrast, the isochrons for site ST270 do not indicate a strong directivity effect. In this case, one would expect waveform properties to be dominated by geometrical spreading and radiation pattern (Fig. 3c) rather than by a directivity effect. The second step in the generation of envelopes is the computation of the response of the medium to a delta function. In the DSM method, the Green’s functions are simplified: they are calculated as Dirac delta functions scaled by

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Figure 2. (a) Acceleration ground motion recorded at Bagnoli station during the 1980 M 6.9 Irpinia earthquake (NS component); (b) Husid plot of the accelogram, low-pass filtered in different frequency-band (cutoff frequencies indicated in the plot); (c) normalized envelopes of the filtered accelerogram.

geometrical spreading (1/R) and radiation pattern (Rh␾) and having a random phase (c), G(t,x,y,z;n,g) ⳱ Rh␾(n,g) cosc d(t ⳮ tr (n,g)) / R,

(1)

where the coordinates (n,g) identify a point on the fault plane in a reference system local to the fault and (x,y,z) identify the site in a global reference system; tr(n,g) is the time of arrival of the rupture front at the point; R is the distance between a point on the fault plane and the site as measured along the ray path. The computation of the radiation pattern term (Fig. 3c), which enhances or decreases the amplitude of wave motion depending on the direction of motion, is based on Aki and Richards (1980; equation 4.80) for farfield S-wave radiation pattern (including both SH and SV terms). The variable c is a random variable that modulates the amplitude of the Green’s functions. The introduction of this random variable means that the contribution to ground motion at a given site due to each isochron is calculated as an incoherent sum of the emitted radiation. It is important to note that the computational mechanics of envelopes requires that the fault be subdivided into several hundreds or thousands of subfaults that are not attributed any physical meaning (they are a simply a computational tracking device). Each of them radiates a Green’s function which, although simplistic, is extremely fast to calculate.

In the third step the envelope E(t) is computed through the following equation: E(t;x,y,z) ⳱

兺n, g

僆l

G(t;x,y,z; n,g) U(n,g) ,

(2)

where the sum is performed over the isochron l, G is the Green function of equation (1), and U is the slip distribution on the fault plane. The envelope is then smoothed and normalized to the unity area to be used in the DSM procedure (Fig. 1). Continuing with the example presented in Figure 3, we turn our attention to the envelopes calculated at three sites located at the same distance from the middle of the fault but for different azimuths (Fig. 3d). The different shapes of the envelopes, as well as the focusing of energy along a preferential direction, are simply the result of the combined effect of radiation pattern and directivity. Rupture complexity at short time and length scales are left to the stochastic component of the methodology (Fig. 1), as the isochron formulation shows that it is the aggregate of these small-scale processes over potentially large fault surfaces that ultimately contribute to the arrival of energy in a given time window at a given site. Finite-Fault Reference Spectrum The reference spectrum of PSSM method is used to introduce a seismological spectral model of the earthquake.

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Figure 3. (a) Contours of rupture time (in seconds) on a vertical strike-slip fault of 20 km length and 10 km width. The envelopes are computed at three sites on a circumference of 15 km radius, centered in the middle of the fault length. The sites are located at 10⬚ (ST10), 190⬚ (ST190) and 270⬚ (ST270) with respect to the strike direction. Contour plots of the (b) isochron curves and (c) radiation pattern values projected on the fault plane and (d) NS and EW synthetic envelopes for ST10, ST270 and ST190. Within the DSM technique, the parameters of the reference spectrum (corner frequency, distance, radiation pattern) are modified to take into account the effects of an extended fault (Fig. 1). The general form to describe the acceleration pointsource-like reference spectrum as a function of the frequency f is: R( f ) ⳱ CA( f )D( f )S( f ) .

(3)

In this equation, C is a constant term accounting for propagation effects, A(f) represents the earthquake source spectrum, D(f) is the attenuation term, and S(f) is the site transfer function. The functional form of A(f) is chosen as an omega square model: A( f ) ⳱ Mo (2pf )2 / [1 Ⳮ ( f / fa)2] ,

(4)

where Mo is seismic moment and fa is the apparent corner frequency which substitutes the standard corner frequency fc used in the PSSM method. The apparent corner frequency

varies from site to site; it is calculated as the inverse of the apparent duration of rupture (Ta) as perceived by the receiver, fa ⳱ 1 / Ta ⳱ 1 / (tcmax ⳮ tmin c ),

(5)

where tmax and tmin are the maximum and minimum isochron c c times from the source to the receiver. In this way, a first approximation of the directivity effect can be introduced in the parameterisation of the source spectrum. Even more, equation (5) can be shown to be equivalent to the corner frequency fc shifted by a directivity function D (Ben-Menhaem, 1961; Aki and Richards, 1980; Joyner, 1991; Bernard et al., 1996) fa ⳱ D fc ,

(6)

with fc ⳱ 1/Tr and D ⳱ [1 ⳮ (Vr /b) cos W]ⳮ1 (Tr is the rupture duration, Vr is the rupture velocity, b is the S-wave velocity and W is the angle between the direction to the receiver and the direction of rupture propagation). This formulation is valid in the far-field approximation and for a


coherent rupture model characterized by uniform slip and constant rupture velocity. The equation (6) can be proved in the two-dimensional case when the isochron velocity c (i.e., the inverse of surface gradient of the arrival-time function) is proportional to the directivity function (Spudich and Frazer, 1984): c ⳱ D Vr .

(7)

Substituting this equation in equation [5] we obtain equation [6]: fa ⳱ 1 / Ta ⳱ c / (TrVr) ⳱ (1 / Tr) (c/Vr) ⳱ fc D The use of the envelope duration to define the apparent corner frequency fa allows us to introduce the information of kinematic rupture description in the spectral model. In this way, the corner frequency is independent of the seismic moment and stress drop, as the duration is a function of the fault dimensions, nucleation point, and rupture velocity. The term accounting for propagation effects in equation [3] is given by: C⳱

Rh␾ FV 4pqb3R

(8)

where F is a free-surface amplification factor usually set to 2, V is a factor to account for partitioning of energy into horizontal components, q is the average density and b the average S-wave velocity. R and Rh␾ represent the average values of the geometrical spreading factor and the radiation pattern, respectively. As we are dealing with an extended fault, we introduce a more elaborate definition of Rh␾ and R. The definitions used in the DSM method involve a spatial average (over the fault) at each isochron time, followed by a temporal average which is weighted by the envelope function itself. Letting F represent Rh␾ or R and using [ ] to denote the mean:

冮F(n,g)dl 冫冮dl

[F(tc)] ⳱

l

l

The line integral is performed over the isochron l corresponding to time tc. Next, the resulting time series of mean values is weighted by the envelope E(t) itself in order to produce the final mean value to be used in the reference spectrum: tmax c

[F] ⳱

tmax c

冮 F(t )E(t )dt 冫冮 E(t )dt c

tmin c

c

c

c

c

tmin c

DSM technique accounts for the total radiation emitted from the extend fault as perceived at each site through the

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parameters of the omega-square model (corner frequency, distance from the fault, and radiation pattern) inferred from the deterministic envelopes. The advantage of this approach is that the spectral content of the time series is not defined by free parameters, which should be calibrated with empirical or theoretical relation. However, the spectral shape is fixed, and it is not derived from the deterministic computation. Even if the omega-square model is the most commonly used to represent the source spectrum, different functional forms could be more appropriate for describing the effects of finite fault. For example, Beresnev and Atkinson (1999) have shown that the total radiation obtained by summing the contribution of each subevent of a large earthquake is characterized by a spectral sag created by the summation process, and corresponds to a functional form with two corner frequencies (Atkinson and Silva, 2000). In the next section, the modeling of two near-source accelerograms recorded during the 1992 Mw 7.3 Landers earthquake is shown, with the aim to highlight the capability and the limits of the DSM technique.

Application to 1992 Landers, California, Earthquake The 28 June, 1992 Mw 7.3 Landers, California, earthquake provided clear examples of forward and backward directivity effects. The earthquake ruptured three vertical fault segments within 30 seconds, with right-lateral strike slip mechanism (Fig. 4). In this case, slip and rupture propagation directions were well aligned, and changes in amplitudes and duration of ground motion recorded at different azimuth have been explained in terms of rupture directivity effects (Herrero, 1994; Herrero and Bernard, 1994; Somerville et al., 1997). In particular, Somerville et al. (1997) have recognized that the differences between the high-amplitude and short-duration recordings at Lucerne (LUC) station and the low amplitudes and long-duration recordings at Joshua Tree (JSH) station are due to directivity effects (Fig. 5). The recordings of both stations are affected by site effects in different frequency ranges. The 1D simulation of the transfer function at JSH shows the amplification in the frequency band 1.5–6 Hz (Fig. 5a). Moreover, a borehole drilled at the LUC site (D. Boore, personal comm., 2002) revealed the presence of a relatively thin upper layer of lowvelocity material that could explain the increased spectral energy observed near 10 Hz (Fig. 5b). To compute the synthetic envelopes, we used a simplified source rupture model based on detailed inversion studies of the 1992 Landers earthquake (Cohee and Beroza, 1994; Wald and Heaton, 1994). The fault model consists of three vertical fault planes (the Johnson Valley and Landers faults, J&L; the Homestead Valley fault, HV; the Emerson and Camp Rock faults, E&C) with the same width but different strike angles and lengths, as listed in Table 1. The regional velocity structure is parameterized in Table 2 (Wald and Heaton, 1994). The rupture model of the Landers earthquake used in

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Figure 4. (a) Map view of Landers fault, with the position of the recording stations used in this study (LUC and JSH) and the epicenter location (star). LUC is located close to the surface rupture and 45 km from the epicenter, facing the rupture front direction, whereas JSH is near the epicenter but in the backward direction of the rupture propagation. (b) Contours of rupture time (in seconds) showing the propagation of the rupture front and (c) contours of spatial slip distribution (in centimetres) on the three faults as defined for this study. The slip was modeled by Wald and Heaton (1994) inverting geodetic survey displacements, near field and regional strong-motions, broadband telesismic waveforms, and surface offsets measurements.

this study is summarized in Figure 4. We assumed that the three faults ruptured independently and nucleated with a delay time in agreement with the observed rupture process (Wald and Heaton, 1994): the mainshock nucleated at 7 km depth on the Johnson Valley Fault; after THV ⳱ 6.9 sec the Homestead Valley fault was triggered, and after TE&C ⳱ 14.3 sec the rupture started on the Emerson and Camp Rock fault. Their nucleation points are in the middle of the intersection segment between two adjacent faults. The rupture propagated radially outward with constant velocity (2.7 km/ sec) and the circular front was perturbed, adding a small stochastic component. The spatial slip distributions on the

three faults (Fig. 4c) are derived from the broadband inversion analysis (Wald and Heaton, 1994). The S-wave synthetic envelopes were first computed separately for the rupture scenario of each fault, and then summed without changing their relative amplitudes (Fig. 6). In fact, event though the simplified Green function cannot be used to compute the observed amplitudes, they preserve the relative values between different scenarios when the same propagation medium is used. In the LUC envelope (Fig. 6b) the contribution of the waves radiated by the three faults is cumulated, and the envelope’s amplitude is controlled mainly by the E&C fault.


233

Figure 5. (a) NS and EW accelerometric components recorded at JSH (left frame). The right frame displays the 1D response computed with Haskell-Thomson method, corresponding to the S-wave velocity profile measured within the Northridge Earthquake project (ROSRINE project, 2001). (b) NS and EW accelerometric components recorded at LUC (left frame). Right frame displays the corresponding spectrogram of the NS component. Note the high-frequency pulses (6 and 10Hz) recorded after the first arrivals (between 14 and 16sec). The recordings are corrected data and delivered by the Consortium of Organizations for Strong-Motion Observation Systems (COSMOS, www.Cosmos-eq.org).

Table 1 Fault Parameters Faults

Mo (⳯1027) (dyne-cm)

Length (km)

Width (km)

Strike (␾⬚)

Dip (d⬚)

Ztop (km)

J&L HV E&C

0.15 0.28 0.31

30 27 36

15 15 15

355 334 320

90 90 90

0 0 0

Table 2 Layered Velocity Model Depth (km)

Vp (km/sec)

Vs (km/sec)

Density (g/cm3)

1.5 4.0 26 32 Bedrock

3.80 5.50 6.20 6.80 8.00

1.98 3.15 3.52 3.83 4.64

2.30 2.60 2.70 2.87 3.50

The JSH site (Fig. 6a), however, perceives the contributions of each fault separately with an evident time delay. In this case the three envelopes have almost the same amplitudes, with the maximum obtained for the J&L fault. The isochrons distribution defines the shape of simulated envelopes: the isochrons are elongated and accelerated towards LUC on the three faults, while they are uniformly spaced in the case of JSH. Moreover, the high slip values on the E&C fault enhance the envelope amplitudes at JSH, making them similar to the amplitudes computed from the J&L fault despite of their different distance from the station. In Figure 7 we compare the synthetic envelopes with the acceleration envelopes of LUC and JSH recordings. The observed time series have been conditioned as follows: (1) we considered the time interval in which 5% and 95% of the energy (Arias’ intensity) had been released; (2) we applied a pass-band filter with a low-cut frequency of 0.1 Hz to avoid contribution coming from near-field terms, and a high-cut of 20 Hz; (3) we tapered and computed the enve-

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Figure 6. Envelopes computed at (a) JSH and (b) LUC sites (horizontal component H ⳱ [(envelopeNS)2 Ⳮ (envelopeEW)2]0.5). Top: overall envelope; bottom: contributions of each single fault, whose nucleation time is shifted in agreement with the observed rupture process: TJ&L ⳱ 0 sec, THV ⳱ 6.9 sec, TE&C ⳱ 14.3 sec. lope of the conditioned signals; and (4) finally, we smoothed both the real and simulated envelopes with a resolution of 1 Hz. Because we are interested on the envelope shape only (the deterministic envelope is normalized before using it to compute the synthetic time series), we multiplied the simulated envelope by a constant factor to reach the maximum observed amplitude. Figure 7 shows the comparison between synthetic and observed envelopes: they have similar shape and duration despite the very simple kinematic source model and the very long simulated duration (20 sec for LUC and 40 sec for JSH). However, some discrepancies can be noted: the first arrival in the simulated motion of the EW component at JSH station doesn’t show the large amplitude of the recorded data, and the second energy pulse is missing in the synthetic envelopes at LUC station. The fit of the simulated and recorded envelopes improves when we apply a high-cut filter of 2 Hz (Fig. 8). In particular, the second large pulse recorded at the LUC site disappears, as it was probably related to both the high-frequency site resonance (Fig. 5b) due to superficial low-velocity material (D. Boore, personal

comm., 2002) and some heterogeneity of the source process. These examples show that the deterministic envelope of the direct body waves reproduces the strong ground motion, both in shape and duration, when site effects or source heterogeneity have been excluded. We performed a simple test to evaluate how a toosimplified representation of the radiation pattern or slip distribution on the fault plane could affect the simulations. The synthetic envelopes at both stations are calculated considering separately the contributions of slip distribution and radiation pattern—that is, with the isochron contribution only (constant slip and radiation pattern), with heterogeneous slip (constant radiation pattern), with variable radiation pattern (constant slip), and with both heterogeneous slip and variable radiation pattern. For the sake of simplicity, this analysis was performed on the envelopes generated by the E&C fault for LUC site and J&L fault for JSH site (Fig. 9). Generally the trend of the envelope is mainly controlled by the isochrons distribution (bottom envelope); the slip and the radiation pattern modulate this basic shape, changing significativley some portion of the envelope. For example, at the JSH station the simulated amplitudes of the first arrivals decrease because the radiation pattern assumes very low values in the portion of the fault around the nucleation point (box, Fig. 9a). At LUC station the coda amplitude of the synthetic envelope is attenuated by the small slip of the north portion of E&C fault, and the second energy pulse disappears when the radiation pattern is added (box, Fig. 9b). To compute the ground motions at LUC and JSH, the synthetic envelopes are normalized and smoothed before windowing the white noise series (steps 1 and 2, Fig. 1). In the following step of the simulation (step 3, Fig. 1), the point-source-like reference spectrum is specified. The parameters necessary to constrain the theoretical spectra have been chosen from the fault that contributes with the most energetic envelope at each site, as shown in Figure 6: the E&C fault for the LUC site and the J&L fault for JSH (Table 3). Conversely, the seismic moment is set equal to the sum of the three seismic moments computed in the inversion performed by Wald and Heaton (1994). The hypothesis is that the low-frequency spectrum, below the corner frequency, is controlled by the whole system of faults, while the highfrequency spectral content is mainly controlled by the most energetic fault, as perceived by the specific site. The reference spectrum computed at JSH includes the 1D transfer function (Fig. 5a). The agreement between synthetic and recorded Fourier amplitudes is satisfactory for both stations at frequencies higher than 1Hz (Fig. 10). It should be noted that DSM is designed to introduce the finite-fault effects in the PSSM method and to reproduce the main feature of the ground motion on the entire frequency band, rather than reproduce the near-source long-period waves where the deterministic techniques are more adequate than the stochastic method. Different causes can explain the mismatch of the simulated ground motion in the low-frequency range (⬍1Hz). First of


235

Figure 7.

Comparison between observed accelerogram envelopes and the overall synthetic envelopes. The NS and EW components, and the horizontal component (H Ⳮ [(envelopeNS)2 Ⳮ (envelopeEW)2]0.5) are displayed. The real signals were filtered with a pass band filter of 0.1–20 Hz.

Figure 8.

The same as Figure 7 with a pass-band filter of 0.1–2 Hz.

all, in DSM the spectral properties of the signals are completely defined by the reference spectrum: source phenomena as large velocity pulses related to the passage of the rupture front or the static offset, cannot be obtained using a simple omega-square model. Recent studies on broadband

simulations (Pitarka et al., 2000) have also shown that a frequency-dependent radiation pattern is more appropriate than a constant value. Moreover, site effects at JSH could be more complex than the 1D response used in our modelling.

236


two horizontal components. At JSH, the simulated peak ground motions agree fairly well with the recorded ones. The underestimate of the observed peak ground velocity reflects the lack of energy of the synthetic seismograms in the low frequency band. These results show that despite of the simplicity of the spectral model, the adopted parameters used to describe the rupture scenario are suitable in the simulation of the main feature of strong-motion data recorded in the near source.

Discussion and Conclusion

Figure 9. Synthetic envelopes computed at (a) JSH site using the J&L faults (EW component); and (b) LUC station using the E&C faults (NS component). From bottom to top: the label NONE indicates constant slip and radiation pattern (i. e., set equal to one); the label ONLY SLIP is for the heterogeneous slip distribution of Figure 4 (constant radiation pattern); the label ONLY RP indicates the envelope with the computed distribution of the radiation pattern only (constant slip); and ALL indicates the envelope obtained considering both radiation term and slip distribution.

Figure 11 shows an example of the time series obtained with the DSM, compared with the recorded accelerations. The synthetic signal is representative of one stochastic simulation, whereas the peak ground motions in Table 3 are the average over 30 synthetics. The comparison with the recorded waveforms is good, especially for the LUC site, where the simulated waveforms of both acceleration and velocity have shapes very similar to the observed data. The model predicts also the peak acceleration values, but underestimates the NS peak velocity value and overestimates the WE value (Table 3). This fact is related to the complex behaviour of finite fault effects: while the deterministic envelopes match the shape of the waveforms on both components, the reference spectrum used in our technique is too simple to simulate the amplitude differences between the

The main goal of the proposed methodology is to include the effects of the rupture propagation on an extended source in the classical point-source stochastic method PSSM (Boore, 1983; 2003a) to predict ground-motion time series. This purpose is achieved through the computation of a deterministic envelope based on the isochron theory; the spectral amplitudes are then scaled with an equivalent pointsource spectrum whose parameters are corrected for the extended fault. There are several advantages of using a deterministic envelope based on the isochron formulation instead of a parametric shaping window: (1) the duration of the envelope is a function of the rupture scenario, such as fault dimensions, rupture nucleation point, and rupture velocity, and it varies from site to site; and (2) the envelope shape depends on the relative position fault-to-receiver and represents how the site perceives the energy release at the source. The modeling at the two testing sites of Lucerne and Joshua Tree shows that the deterministic envelopes reproduce the main features of the observed recordings, such as the directivity effects associated with the rupture propagation. The Lucerne simulation (directivity site) is dominated by the cumulative effect of the energy radiated from the three faults. The result is a synthetic envelope with short duration and high amplitude. On the other hand, the Joshua Tree simulation is affected by the spreading out of energetic arrivals radiated from the faults. The synthetic envelope is characterized by long duration and low amplitude. These effects are controlled by the fault geometry and by the rupture propagation, rather than by the details of the rupture process (heterogeneity on the slip distribution and/or rupture times) as shown by the comparison with the observed envelopes filtered at different frequency bands. Moreover, the shape of the envelope is mainly controlled by the isochrones distribution; the slip and the radiation pattern modulate this basic shape, changing significantly some of its parts. The deterministic computation allows us to define the parameters to be included in the theoretical spectrum: the apparent corner frequency fa, calculated as the inverse of the envelope duration; and the two averaged parameters, distance R and radiation pattern Rh␾. These parameters are modified by the finite-fault effects, and they control amplitudes and frequency content of the synthetic signals. The corner frequency has a first-order effect on the spectrum: it defines

237


Table 3 Spectral Parameters, and Synthetic and Recorded Peak Ground Motions Site

Fault

Mo (dyne-cm)

fa (Hz)

JSH

J&L

7.4e27

0.073

LUC

E&C

7.4e27

0.071

Rh␾

R (km)

PGAsynt (g)

PGArec (g)

NS EW 具H典

0.36 0.43 0.53

19.2 23.0 21.6

0.19 Ⳳ 0.03 0.18 Ⳳ 0.02 0.25 Ⳳ 0.03

0.27 0.28

18 Ⳳ 3 18 Ⳳ 3 24 Ⳳ 4

27.3 42.9

NS EW 具H典

0.45 0.49 0.68

10.7 11.2 10.8

0.65 Ⳳ 0.09 0.75 Ⳳ 0.19 0.94 Ⳳ 0.25

0.69 0.66

61 Ⳳ 23 62 Ⳳ 21 93 Ⳳ 39

30.5 139.0

PGVsynt (cm/sec)

PGVrec (cm/sec)

Figure 10.

Comparison between observed and synthetic amplitude Fourier transform of NS and EW components at JSH and LUC sites; the horizontal composition of the spectra (H ⳱ [(spectrumNS)2 Ⳮ (spectrumEW)2]0.5) is also displayed. Each synthetic spectrum is an arithmetic average of the Fourier spectra of 30 simulated signals. Both synthetic and recorded accelerograms were low-pass filtered with a 20 Hz corner frequency.

the low-frequency content and scales the high-frequency spectral amplitude (acceleration spectral level) as the square of its value. Several features caused by rupture directivity and affecting ground-motion time series have been studied and included in theoretical models (Bernard et al., 1996) or empirical regressions (Boatwright and Seekins, 1997; Somerville et al., 1997). These approaches include directivity effects using some functional form compatible with the well-known Ben-Menahem (1961) directivity function, which depends on the angle W between the rupture direction and the direction to the receiver. This angle is constant in the point-source approximation and in this case the modification of the corner frequency with the directivity function causes an unrealistic overestimate of the spectral amplitudes for directive sites (Bernard et al., 1996; Berge et al., 1998).

Indeed, when the rupture propagates on an extended fault, the angle W changes in space and time, and in this case a rms directivity has to be considered. We tested different averages, such as arithmetic or quadratic mean, average weighted by the subfault distance (Boatwright and Seekins, 1997), or average computed along the isochron and weighted by the envelope, as proposed in this work, for computing the fault distance and the radiation pattern. Each average gives different results: the directivity factor varies from 30% to 70% of the point-source value for a directive site, and from 60% to 200% for an anti-directive site. On the other hand, the apparent corner frequency of the DSM method, computed directly from the isochron arrival times, includes the generalized directivity function without dealing with an average operator, and it is uniquely defined for a given rupture scenario and source-to-receiver geometry.

238


Figure 11. Synthetic and observed waveforms comparison of NS and EW components at JSH and LUC sites (acceleration and velocity). For completeness, we included the time series obtained using the horizontal envelopes (H) of Figure 6. The simulated signal is representative of one stochastic simulation.

The finite dimensions of the fault also affect the definition of the fault distance R and the radiation pattern Rh␾ to be used in the DSM spectrum. These parameters are averaged over the fault plane, taking into account each subfault contribution as perceived by the receiver. At the same position from the fault, the geometrical spreading and the radiation pattern have a second-order effect on the ground motion, compared to the directivity. Moreover, we find that their mean values are weakly dependent on the average operator. The calibration of the synthetic amplitudes using the point-source-like reference spectrum is a good choice to reproduce observed peak ground accelerations when the recordings are not affected by important site or propagation effects. When a site effect is recognized, it can be included in the frequency domain using a transfer function, without modifying the deterministic envelope (as for JSH). The discrepancy with the real spectra in the low frequency range can be ascribed either to source process phe-

nomena or site effects. Records of recent earthquakes (1995 Kobe, 1999 Izmit, and 1999 Chi-Chi) show that the highvelocity pulses are due to constructive interference of longperiod waves associated with the propagating rupture; such ground motion cannot be simulated using stochastic methods in time domain, and they are not considered in the parameterisation of the point-source-like reference spectrum of DSM. Moreover, DSM simulations are computed for direct body waves in the far-field approximation. These simplifications limit the applicability of the method when effects due to near-field terms, superficial, diffracted, or reflected waves are expected to be significant. The proposed technique can reproduce the main characteristic of strong-motion recordings and can be implemented using only a limited number of parameters to describe the source (dimension and geometry), the propagation medium (wave velocities and layers), and the site (transfer function). These characteristics are important for a methodology aimed to simulate ground shaking scenarios. Seis-


mic parameters for engineering purposes, such as PGA and PGV, response spectra, Arias’ intensity, and duration of strong ground shaking, for example, can be obtained for a large number of synthetics. Strong-motion envelopes can be easily and quickly generated from a large number of plausible rupture scenarios of an extended fault. Therefore, the uncertainty associated with the ground motion predictions reflects the different physical assumptions used for the scenarios.

Acknowledgments The authors are grateful to David Boore for the valuable comments and suggestions that helped to improve the manuscript. An anonymous reviewer and the associate editor Igor Beresnev are also acknowledged. The original ideas that motivated this work were developed when two authors were working at ISMES Spa, supported by ENEL Spa. We thank both these companies and Raniero Berardi, who promoted our collaboration in the early stages of this research project. This study was partially supported by GNDT project 2000–2003 “Development and comparison among methodologies for the evaluation of seismic hazard in seismogenic areas: application to the Central and Southern Apennines,” funded by the Italian Civil Protection.

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