Pure appl. geophys. 158 (2001) 647±665 0033 ± 4553/01/040647 ± 19 $ 1.50 + 0.20/0

Ó BirkhaÈuser Verlag, Basel, 2001

Pure and Applied Geophysics

Focal Mechanisms of Weak Earthquakes from Amplitude Spectra and Polarities JIRÏIÂ ZAHRADNIÂK,1 JAROMIÂR JANSKYÂ1 and KONSTANTINA PAPATSIMPA2

Abstract Ð The ASPO method (Amplitude Spectra and POlarities) for the focal-mechanism retrieval from relatively weak events is based on a widely available instrumental setup: A few broadband stations within a denser short-period network. Collectively all stations provide the epicenter location. Complete records are taken from three-component broadband stations, without selecting a particular wave type, or picking amplitudes. It makes the method suitable for automated data processing, and enables studies of the interference crustal phases. Only the amplitude spectra are inverted. This is a robust feature which makes the method insensitive to any timing problems (such as those due to uncertain origin time, or due to technical failures). The ®rst-motion polarities serve as an additional constraint of the amplitude-spectra inversion; only few (clear) polarities are taken from the nearest stations, where they mostly belong to direct P waves. The method seeks ®ve parameters: The focal depth, scalar moment, strike, dip, and rake. Green's function, automatically including possible near-®eld e€ects and interference (e.g., surface) waves, is calculated by the discrete wavenumber method. ASPO works below the corner frequency, and the time function is not being retrieved. This feature not only minimizes the number of the inverted parameters, but also speeds up the calculation, because the lower the frequency, the faster the discrete wavenumber run. Instead of an exceedingly slow 5-parameter grid search, the inversion is organized in two steps: (i) the depth and moment determination with a coarse grid search of the strike, dip and rake, and (ii) a ®ne grid search of the three source angles. Uncertainty of the best-®tting solution is assessed from the minimum error value and from the scatter of the nodal lines (and/or P and T axes) between min and min + 10%. The method was tested on the clustered M » 3.5 earthquakes recorded by a temporary network of three CMG3-T broadband stations in western Corinth Gulf. A fundamental problem is that the broadband stations su€er systematically from event-induced instabilities at horizontal components if earthquakes of the studied magnitudes occur at short distances, 10±30 km. Therefore, the ASPO method could not be applied below 0.1 Hz. As such, the results are sensitive with respect to unknown crustal structure details, and the focal mechanisms remain rather uncertain (minimum error higher than 0.34). Compared to synthetic tests with perturbed data, in which the error is lower than 0.2, it is concluded that the crustal model needs further improvement. Key words: Focal mechanisms, weak earthquakes, broadband stations, Mediterranean, Greece, Corinth Gulf.

1

Charles University, Faculty of Mathematics and Physics, V Holesovickach 2, 180 00 Prague, Czech Republic. E-mail: [email protected]€.cuni.cz 2 University of Patras, Seismological Laboratory, 261 10 Rio, Greece.

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Introduction Corinth Gulf, separating continental Greece and the Peloponnese Peninsula, is a rift belonging to the seismically most active regions of the Mediterranean (MAKROPOULOS et al., 1989; PAPAZACHOS and PAPAZACHOU, 1997). Focal mechanisms of the stronger Gulf events, M > 4.5±5.0, have been relatively well known (AMBRASEYS and JACKSON, 1990; TSELENTIS et al., 1996a; BERNARD et al., 1997; ARVIDSSON and EKSTROÈM, 1998). However, weaker events also must be studied in order to understand the seismotectonic details of that region (MELIS et al., 1989, 1995; HATZFELD et al., 1990; GIBOWICZ et al., 1999), and to correctly apply weak-motion records as empirical Green's functions (PLICKA and ZAHRADNIÂK, 1998; PLICKA et al., 1998a,b). Therefore, this paper concentrates on methodical aspects of the focal-mechanism inversion from weak earthquakes, and gives a few examples of the applications of the suggested method in the Corinth Gulf. The focal-mechanism determination for weaker events (magnitude lower than 4.0) is not an easy task. Events are recorded at a few stations only, thus methods based purely on ®rst-motion polarities are disabled. The propagation paths are short, thus signi®cantly a€ected by unknown small-scale crustal heterogeneities. This calls for methods based on low frequencies (e.g., FAN and WALLACE, 1991; JONES et al., 1993), however weak events are not very rich in them. Thus the selection of a proper instrument and an appropriate frequency window is critical. Time-domain inversions, or methods employing complex spectra, are vulnerable to the location inaccuracies unless special devices are used, such as arti®cial time shifts (SÏIÂLENY et al., 1996; SÏIÂLENY and VAVRYCÏUK, 2000). Methods retrieving the complete moment tensor in the poorly known structures often provide non-realistic deviations from the double couple, unless applying the double-couple constraint (GIBOWICZ et al., 1999). When inverting simultaneously for the focal mechanism and the source time function, special methods must be applied to prevent the undesired mapping of the unknown near-source heterogeneities into the source complexities (SÏIÂLENY et al., 1992; JECHUMTAÂLOVA and SÏIÂLENYÂ, 1998). Similar problems arise from the uncertain focal depth, if taken from the kinematic location only. Instead, this paper suggests inversion of the source depth, scalar moment, strike, dip, and rake (i.e., a double-couple constrained inversion, without time function), made robust by employing amplitude spectra of complete waveforms below the corner frequency at a small number of broadband stations. The amplitude inversion is complemented by a few clear ®rst-motion polarities. The best-®tting solutions are validated by comparing the observed and synthetic seismograms. Scatter of the solutions whose error falls within a prescribed range serves as a measure of the uncertainty. It is believed that the focal mechanisms resulting from this method may yield reliable time functions as a following research task. The acronym for the method will be ASPO, denoting amplitude spectra and polarities. The combination of the amplitude and polarity data (cf., also

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RONGWALDSSON and SLUNGA, 1993; KRAVANJA et al., 1999) represents an attempt to partly bridge the existing gap between the short-period kinematic and broadband dynamic solutions. The goal to study the complete records without separation of individual waves, including the interference crustal waves and near-®eld e€ects, implies that the velocity spectra of the complete records will have no simply de®ned slope below the corner frequency, such as f1. Therefore, full elastodynamic equations must be solved. We use the discrete-wavenumber method, DW, for point sources in 1-D models (BOUCHON, 1981; KENNETT and KERRY, 1979; COUTANT, 1989).

Method The focal-mechanism method of this paper is based on modeling the observed displacement amplitude spectra. The modeling employs a ®xed epicenter, although it varies the focal depth. The other focal parameters to be found are the scalar moment, strike, dip, and rake. The synthetic amplitude spectra are obtained as follows: The Green's function is calculated for a trial depth. For trial values of the scalar moment, strike, dip, and rake, the moment tensor is calculated, whose time function is a step. (In fact, working below the corner frequency, the actual time function is unimportant.) The moment tensor and Green's tensor are multiplied in the complex spectral domain, and the modulus is taken. A correction is applied, compensating the arti®cial attenuation employed in the DW method as a regularization and anti-alias operation. The observed spectrum is calculated and saved, thus it can be inverted repeatedly with di€erent options of the norm, stations, components, frequencies, depths, and the other grid-search parameters. In this paper the observed and synthetic amplitude spectra are compared in a modi®ed L1 norm, hereafter called L1*. The error (or mis®t) function in the L1* norm is de®ned as follows: ntest P

error ˆ

iˆ1

jobsi syni j max…jobsi j;jsyni j†

ntest

:

The summation is performed over all stations (nstat), all components (ncomp), and all considered frequencies (nfreq): i ˆ 1; 2; . . . ntest, where ntest ˆ nstat*ncomp*nfreq. The factor max (¼) partly reduces the undesired biasing e€ect of the largest amplitudes at the nearest station (-s). Moreover, it normalizes the maximum possible error to 1. It is important that the amplitude spectra must not be smoothed prior to modeling. A detailed representation of the individual spectral peaks and troughs is a

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prerequisite for achieving results comparable to the time-domain modeling, such as, for example, correct amplitude ratios of the P- and S-wave groups. Instead of a lengthy 5-parameter search, the procedure is divided in two steps: (Step 1) A coarse search over strike, dip, and rake angles with a few trial focal depths (5±10 values) and scalar moments (20±30 values). The minimum error at each moment is inspected as a function of depth. (Step 2) A ®ne grid search of the strike, dip, and rake is performed for the best depth and moment found in Step 1, and also for at least two neighboring depths. Finally, the error function for the optimum depth is plotted against the sequential trial number. By the trial we mean a pair of the strike and dip values, complemented by that rake for which the error is minimal at the given strike-dip pair. See example in Figure 1 of the next section. The error graph is oscillatory, because the formal succession of the trials (two loops over strike and dip) puts the physically close solutions formally apart from each other. Free of this limitation are 3-D error plots which, however, are not needed in this paper. It is enough to vary the rake from 0 to 180 (not )180 to 180), since the inversion of the amplitude spectra makes no di€erence between the mechanisms with a rake R,

Figure 1 Synthetic test. One `earthquake' is inverted in three di€erent modes: (top) with the correct depth and moment, (middle) with the incorrect depth and correct moment, and (bottom) with the incorrect depth and incorrect moment. The line graph shows the error function of the amplitude-spectrum inversion without considering polarities. The diamonds denote solutions whose error ranges between minimum and minimum +10%, and 3 prescribed polarities are satis®ed. All these ``acceptable'' solutions are also presented by the nodal lines and P-T axes. The test helps to evaluate the quality of the real-data inversions in Figures 5 and 6.

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and R-180. These two mechanisms have waveforms of the opposite signs, but the amplitude spectra and errors are identical. This fact saves the computing time needed for the search, but requires independent use of at least one polarity to distinguish between R and R-180. The solutions for R and R-180 should not be mixed with the conjugated solutions. The conjugated solutions correspond to twin triplets of the strike-dip-rake, describing the two nodal planes. Although both triplets represent the same physical mechanism, the search gives them both (see, for example, the two minima around the trial numbers 200 and 1000 in Fig. 1). As a next step, ray-theoretical ®rst-motion polarities are calculated for all strikedip-rake triplets of the ®ne grid search, and are compared to the observed polarities. Since only rakes 0±180 are grid-searched, we consider a fault-plane solution successful not only when all observed polarities are satis®ed, but also when all polarities are opposite. In the latter case the trial value of the rake transforms from R into R-180. If the polarities are satis®ed for more than one rake value of a given trial number (a strike-dip pair), only the minimum-error rake is saved. That particular error value is plotted by a diamond symbol in the error-function line graph. Note that for a given strike-dip pair the minimum error satisfying polarities can be larger than the minimum amplitude error (diamond above the solid line). On the other hand, it cannot be lower (diamonds never below the solid line). See examples in Figure 1. It may perhaps appear surprising that the polarity agreement is tested after the amplitude inversion, and not vice versa. However, succession like that is advantageous for two reasons: (i) Once the amplitude checking is done, results can be saved and repeatedly tested for di€erent subsets of the polarities. (ii) Relation between the amplitude-satisfactory solutions and the polarity-satisfactory solutions serves as an important indication of the internal consistency of the model. The best possible condition exists if some diamonds coincide with the minima of the line graph (Fig. 1), thus showing that the waveform spectra and ®rst-motion polarities are both explained by the adopted crustal model. To estimate the uncertainty of the solution, all polarity satisfying solutions whose amplitude error falls within a given range are inspected by plotting of their nodal lines, and/or the P and T axes (equal-area lower-hemisphere projection, program FPPLOT by REASENBERG and OPPENHEIMER, 1985). The ®nal validation of the best-®tting solutions is accomplished by comparing the observed and synthetic amplitude spectra and seismograms (see examples in the next sections). Technical Remark: Variation of the error with moment and depth is datadependent. If an earthquake displays a dramatic change, it is useful to repeat the coarse search in a `rotated mode'. Let us assume that we were searching the strike from 0 to 360, with 20 degrees increment. By the rotated search we mean the variation with the same increment of 20, but starting at 10. The two ``rotated''

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searches are nearly as useful as a single one with increment 10, however if applied to three angles they need considerably less operations (roughly 2 ´ M3 compared to (2M)3).

Synthetic Test A synthetic test was performed to understand the resolution of the ASPO method. We de®ned a theoretical event resembling to the extent possible one earthquake of the next chapters (event 09251934). We prescribed the source depth 7 km, scalar moment Mo ˆ 2.0 e14 Nm, strike 310, dip 70, and rake )40 degrees. The synthetic seismograms were generated and then processed exactly as real data in Step 2, as regards the station distribution, crustal model, number of components, frequency range, grid-search parameters, etc. Also three polarities corresponding to the prescribed mechanism were included in the inversion. The synthetic data were processed in three modes (Fig. 1): (a) Inversion with correct Green's function, i.e., that for 7 km depth, and correct moment 2.0 e14 Nm, (b) inversion with incorrect depth (6 km) and correct moment, (c) inversion with incorrect depth (6 km) and incorrect moment (2.5 e14 Nm). Case (a) yields the correct strike, dip and rake in the form of a very deep error minimum (=0.0128). Case (b) is also good, the best-®tting strike, dip and rake di€er from the correct values only by 5 degrees. The minimum is still well developed, but not deeper than 0.2039. Case (c) alters the fault plane solution more than (b), and the minimum becomes less distinct and shallower (0.2909). Advancing from (a) to (c), we obtain a larger variability of the solutions close to its optimum. Choosing, for example, a range from the minimum error to minimum plus 10%, we ®nd that the ideal case (a) yields just a single (=highly reliable, well resolved) solution, (b) provides a very compact family of the solutions, and (c) gives a larger variation. Note, however, that all considered perturbations still kept the correct structural model. Therefore, with real data, one can hardly expect errors lower than 0.3 and the scatter of the nodal lines and P, T axes smaller than in the case (c) of Figure 1.

Crustal Model Ironically, our wish to keep the method simple by studying complete seismograms disables use of simple crustal models. Indeed, in a standard crustal model routinely used for locating earthquakes in western Greece, the ``complete-waveform'' synthetics provided by the DW method are in fact incomplete, or oversimpli®ed, and much shorter than the records. Two possible solutions to this problem are as follows: (i) A simple crustal model can be kept, and a cut-and-taper applied to the

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records; then only short record segments allowing the DW modeling are subjected to the inversion. However, the cut-and-taper operation is delicate, destroying the robustness of the method, thus it should be avoided wherever possible. (ii) A more realistic crustal model must be provided. In this paper we employ one ad hoc crustal model for the studied region and events which yield the DW synthetics of a realistic duration. The fortunate experience from the Corinth Gulf is that the low-velocity subsurface crustal layers, mostly responsible for the long durations, are strongly constrained by surface waves at epicentral distances exceeding 20 km. That was the main guideline for the trial-anderror modi®cation of the previously available models. The original simple crustal model, used in the studied region for many years in routine locations (TSELENTIS et al., 1996a), here called M1, has its topmost layer 5-km thick, with Vp ˆ 5.7 km/s; see Table 1. A recently introduced model MN2, including strati®cation in the topmost 5 kilometers, and subsurface velocities as low as Vp ˆ 1.42 km/s (TSELENTIS et al., 1996b; TSELENTIS, 1998), proved to explain large ground-motion durations. The Vp/Vs ratio in that model is 1.78 everywhere. The quality factor is 300 in the crust (TSELENTIS, 1993), and 1000 below Moho, respectively. The preferred model of this paper, MN6, is a minor modi®cation of MN2, being simpler below 5 km (Table 2). With the preferred crustal model we do not need any record cut at two broadband stations. However, as shown below, the third station possessing a signi®cant site e€ect still requires a partial cut before computing and inverting the amplitude spectra. Table 1 The original crustal model M1 Depth (km)

Vp (km/s)

0 5 18 39

5.7 6.0 6.4 7.9

Table 2 The preferred crustal model MN6 Depth (km) 0 0.5 1 2 5 39

Vp (km/s) 1.42 2.67 4.45 5.7 6.4 7.9

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Once the crustal model includes low subsurface velocities, it may contain relatively strong velocity contrasts. For example, in our preferred model this occurs at the depth of 5 km. As a result, shallow events located above the 5-km discontinuity have their ®rst arrivals at longer distances formally interpreted as the 5-km interface (``head'') waves, hereafter denoted Pn5, whose take-o€ angles are relatively small, say 60 degrees. Now let us imagine that the source moves slightly below 5 km. Then the ®rst arrivals are direct P waves, whose take-o€ angles are about 90 degrees, signi®cantly di€erent from 60. Unless having local travel-time curves with a clear Pn5 phase, employment of such a model for the polarity projection onto the focal sphere is too formal and dangerous. Equally dangerous is to interpret all ®rst motions as direct waves, because some of them can be head waves connected to intercrustal discontinuities missing in our model. For a similar discussion regarding the uncertain nature of the ®rst arrivals, see SHEARER (1998). Problematic (model-dependent) projections of the stations onto the focal sphere are common to most papers dealing with the focal mechanisms. To reduce the complications like that we will employ only clear ®rst-motion polarities at short epicentral distances where almost certainly they correspond to direct P waves. At the same time, the source depth will vary during the amplitudespectra inversion (as, for example, in JONES and HELMBERGER, 1998). Then the same trial depth will be consistently used for computing Green's function as well as for projecting polarities onto the focal sphere.

Earthquake Data The velocity records and polarities are taken from three CMG-3T Guralp broadband stations, Table 3 and Figure 2, operating in the western Corinth Gulf since 1997 (ZAHRADNIÂK and TSELENTIS, 1999). The epicenter location is provided by the telemetered short-period PATNET network, covering western Greece since the 80s (TSELENTIS et al., 1996a; MELIS et al., 1995), supplemented as well by the readings from the broadband stations. We are interested in earthquakes recorded by all three BB stations with a good signal-to-noise ratio, and located reliably. Since the BB stations are distributed at the

Table 3 The broadband-station coordinates Station

Lon. (deg.)

Lat. (deg.)

CLAU EGIO KALI

38.198 38.223 38.413

21.772 22.037 22.057

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Figure 2 Broadband network of the Charles Univ. and the Patras Univ. in Corinth Gulf. Stations CLAU, EGIO and KALI are denoted by triangles. Epicenters of the earthquake cluster of September 1998 are designated by crosses. Focal mechanisms, obtained in this paper, are shown by the beach balls. For the reliability of the solutions, see also Figure 6 and Table 5.

periphery of PATNET, only earthquakes of magnitude >2.5±3.0 (the `duration' magnitude, TSELENTIS, 1996a) satisfy these requirements. A cluster of ®ve magnitude 3.4±3.7 events (Table 4 and Fig. 2) has been selected for testing the focal-mechanism inversion suggested in this paper. The events occurred halfway between the two BB stations, EGIO and KALI (epicentral distance of about 10 km each). The notable di€erence between the EGIO and KALI records (Fig. 3) is due to a strong site e€ect at EGIO (JANSKYÂ, 1999). The third BB station, CLAU, had a distance of about 28 km.

Table 4 The epicenter location and magnitude of the studied events #

Event code

1 2 3 4 5

09162305 09171728 09251934 09260740 09261150

Orig.time (h-m-s) 23 17 19 07 11

06 29 35 41 51

13.10 06.61 13.25 16.02 20.11

Lon. (deg.)

Lat. (deg.)

M

38.319 38.328 38.309 38.313 38.311

22.047 22.042 22.081 22.066 22.078

3.4 3.5 3.4 3.7 3.6

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Figure 3 The broadband un®ltered velocity records of a M ˆ 3.5 event (09171728). Note a signi®cant complexity of the records, in particular the site e€ect at EGIO station (whose epicentral distance is the same as that of KALI station). The disturbed baseline due to an event-induced instability at the CLAU E-component can also be seen.

We use complete records at CLAU and KALI (30 and 20 seconds respectively, starting at the origin time), while only 8 seconds are inverted at EGIO due to the site e€ect. The initial goal was to work in the lowest available frequency range, believing that the uncertainty of the crustal model has a minimal e€ect there. However, with CMG3T and the event selection described above, the actually usable range is only f > 0.1 Hz. It is because at lower frequencies the horizontal components su€er from signal-induced instabilities. Although still below the clipping level, local events rich in relatively high frequencies drive the pendulum into a sudden motion (``jump''), not properly compensated by the force-balance system, hence the velocity record produces a disturbance (roughly 2 minutes long) similar to the instrument-calibration response. The earthquake record becomes superimposed on such a perturbed baseline. For details of this phenomenon, called ``mice,'' see ZAHRADNIÂK and TSELENTIS (1999). The beginning of a ``mouse'' is well visible at CLAU-E record

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(Figure 3), however smaller disturbances also exist in the other stations. Thus the EW component at CLAU was excluded from the present study, and the amplitude spectra of the remaining components were inverted above 0.1 Hz (except KALI-E, which was used above 0.2 Hz). The upper limit of the inversion, below the corner frequencies, was set at 1 Hz. The observed and synthetic displacement spectra were sampled with the increment of 1./81.92 ˆ 0.0122 Hz, small enough to represent detailed variations. All epicenter determinations, polarity projections onto the focal sphere, and the amplitude-spectra inversions were performed in crustal model MN6 (Table 2). Only three polarities were used, just those from the BB instruments which, for the selected events, represent the nearest stations. For all ®ve events the ®rst P arrivals on Z component are compressional. For each event the focal depth varied from 4 to 11 km, with an increment of 1 km. An example of event 09171728 is given in Figure 4 where, at each trial depth, the moment varied from 2 to 5 e14 Nm. The coarse grid search tested the strike from

Figure 4 Results of the depth and moment determination for the 09171728 event. The amplitude spectra are inverted by the coarse strike-dip-rake search. The minimum error corresponding to each trial moment is shown for the trial depths of 4 to 11 km (see graph labels). The present example prefers the depth of 7 km and moment of 3.5 e14 Nm, for which a subsequent ®ne grid search of the strike, dip, and rake was performed in Figure 5.

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10 to 360, dip from 10 to 90, and rake from 0 to 180, with increments of 20, 10 and 20 degrees, respectively. The smallest errors corresponding to each moment indicated the optimum depth of 7 km and moment 3.5 e14 Nm, in Figure 4. The resulting depths and moments for all ®ve events are in Table 5. Using the optimum depths and moments, the ®ne search was performed to investigate the strike from 5 to 360, dip from 5 to 90, and rake from 5 to 180 with increments of 5 degrees. For the results of the amplitude inversion, constrained by the three polarities, see Figure 5 and Table 5. As seen in Figure 5, two earthquakes (09251934, 09261150) have larger minimum errors and less distinct minima than the remaining three. This is even better expressed in Figure 6, where all acceptable solutions are plotted, thus measuring the uncertainty. The earthquakes 09171728 and 09260740 yield the most reliable solutions, in particular the well constrained, nearly NS-oriented, horizontal T axis. The best-®tting solutions are also presented in the map (Fig. 2). Two solutions (#1 ˆ 09162305 and #2 ˆ 09171728) are nearly identical normal-faulting mechanisms. The solution #4 ˆ 09260740 is also normal fault, similar to the previous two, but with a 180-degree change of the strike. The two least reliable events (#3 ˆ 09251934, #5 ˆ 09261150) have their best-®tting solutions close to the reliable solution #4. We cannot go into a more detailed discussion, since the number of the studied events is small, and in this paper they mainly serve as methodical examples. Although the duration magnitudes are close to each other (3.4±3.7), the scalar moments range from 0.9 e14 to 1.6 e15 Nm (Table 5). It is possible that the largest moment resulted from multiplicity of the 09261150 event (where a ``foreshock'' was indicated at 1.3 s before the ®rst arrival), which would also partially explain its poor resolution. Finally, the observed and synthetic data are compared for earthquake 09171728 (the lowest error of this paper, 0.3403) in Figures 7 and 8. The agreement is global, Table 5 The focal depth and the best-®tting mechanism of the studied events #

Event code

Depth (km)

Moment (Nm)

Strike, dip, rake (deg.)

Paz, Pi (deg.)

Taz, Ti (deg.)

Error

1

09162305

7

0.9 e14

97, 60

3, 2

0.3429

2

09171728

7

3.5 e14

89, 58

183, 2

0.3403

3

09251934

9

2.2 e14

273, 33

27, 33

0.3786

4

09260740

7

6.3 e14

277, 44

13, 7

0.3418

5

09261150

9

1.6 e15

120, 50, )50 247, 54±127 120, 55, )50 244, 51, )133 150, 90, 50 60, 40, 180 65, 55, )150 317, 66, )39 60, 45, 175 154, 86, 45

278, 27

27, 33

0.3546

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Figure 5 Results of the ®ne strike-dip-rake grid search for ®ve earthquakes. Fixed depths and moments, found in the previous coarse search, are used. The line graph delineates the error function of the amplitude-spectrum inversion without considering polarities. The diamonds denote solutions whose error ranges between minimum and minimum +10%, and 3 polarities are satis®ed. The minimum error and the best-®tting mechanisms are given in the legend.

i.e., the synthetics well explain the relative di€erences among the individual stations and components. A very good spectral ®t can be seen only in KALI-Z, up to 0.6 Hz. It is also noteworthy to mention that the CLAU-E spectrum is not modeled worse than the others, although the observed CLAU-E component was not used in the inversion. Well modeled waveforms are KALI-N and EGIO-N. Surface waves at later arrivals in CLAU are partially modeled, too. However, many details of the spectra and waveforms remain unexplained, as also expressed by the error value, signi®cantly larger than in the synthetic tests. This implies that the crustal model needs further improvement. Our experience is that the results do not change practically if we invert the velocity or displacement spectra. This is due to the relatively narrow frequency range studied in this paper, and due to the used norm L1*. The reason why we have ®nally selected displacement is the presentation of the time-domain results. In fact, working

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Figure 6 The nodal lines and P,T axes for ®ve earthquakes. All polarity and amplitude satisfying solutions within the acceptable error limits (Fig. 5) are denoted by light lines and symbols. The best-®tting solution is given in bold, and the error values appear below the graphs. For one event (09162305) the three compressive polarities are also shown in the right top corner.

below the corner frequency, presenting velocity would be misleading, since only the ®ts/mis®ts at the highest studied frequency then dominate.

Discussion and Conclusion A robust method for the focal-mechanism determination of relatively weak events has been suggested. It is assumed that the studied events have reliable epicenters, e.g., from a local short-period network, and are recorded at a few three-component broadband stations. The complete records are used to invert their amplitude spectra, with a simultaneous control by a few clear ®rst-motion polarities at the nearest stations. The acronym is ASPO. Let us compare the ASPO method with other methods. It may seem that if a weak earthquake is located successfully, its mechanism can be retrieved solely from

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Figure 7 Comparison between the observed (thick line) and synthetic (thin line) displacement amplitude spectra for the best focal mechanism of event 09171728. Only the range from 0.1 to 1.0 Hz, subjected to the inversion, is shown. The CLAU-E data were excluded from the inversion for the instrumental problems, but are ®tted relatively well for f > 0.3 Hz. The KALI-E data were inverted from f > 0.2 because of similar (but less serious) instrumental problems as in CLAU-E.

polarities. This is not always the case for several reasons: (i) Some polarities are unclear due to noise, (ii) all polarities cannot be satis®ed simultaneously, (iii) the polarities can be satis®ed, however they provide a broad family of the fault-plane solutions without any preference (except the statistical one), (iv) the polarities may yield a wrong solution if imprecisely projected on the focal sphere due to uncertain focal depth obtained from the purely kinematic location. The ASPO method removes all the mentioned problems: only very few clear direct-P polarities are used, and the optimum depth and the best-®tting fault plane solutions are inferred from the amplitude-spectra mis®t function. Compared to the time-domain inversion of complete waveforms, the ASPO method is faster and more robust: Computing time is saved because one FFT at every grid-search trial is eliminated. Robustness comes from insensitivity to the timing problems, such as those due to uncertain origin time, the GPS malfunction, etc. Elimination of the need to identify selected wave types and/or to pick the amplitudes from the records is another advantage of the ASPO method, making it

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Figure 8 Comparison between the observed (thick line) and synthetic (thin line) displacement time histories for the best focal mechanism of event 09171728. The seismograms are bandpass ®ltered between 0.1 and 1.0 Hz. The time scale reads zero at the origin time minus 10 seconds. The horizontal bar above EGIO-Z shows the modeled part of that record. The original (unused) resonant part of EGIO-Z is shown by the dotted line. Although similar in EGIO-N and E, the resonance is not shown there to preserve the clarity of the modeled part.

suitable for automation. Moreover, using complete waveforms, ASPO attempts to reduce the unmodeled part of the record, in particular compared to the inversions based solely on peak amplitudes. On the other hand, the use of complete records also carries diculties: We need a relatively good crustal model in which synthetics explain the entire duration of the records. This means that, wherever possible, stations with a strong site e€ect should be excluded (mainly those with long duration due to resonances). Another alternative is to carefully shorten the observed waveforms by removing the problematic later arrivals. The ASPO method not only provides the best-®tting mechanisms, but their reliability can be evaluated by inspecting families of the nodal lines and P-T axes falling into an acceptable error range (minimum error plus 10%). We have applied the method in Corinth Gulf, and actually have obtained di€erent uncertainties for di€erent events in spite of their tight clustering in space and time. Therefore, an interesting unsolved question remaining for future studies is why certain earthquakes are resolved better than others.

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Mis®t in this paper was measured by an L1* norm, slightly modi®ed from L1 to reduce bias due to the largest amplitudes. As such, the error is theoretically between 0 and 1. As indicated by the synthetic tests with perturbed parameters (e.g., inaccurate Green's function), real data can hardly provide error lower than 0.2. Indeed, the lowest error obtained in our applications was 0.34. This means that the crustal model adopted in this paper still needs further improvements. A recent study based on the Corinth Gulf stations and surface wave dispersion from regional earthquakes will help in this respect (NOVOTNY et al., 2000). It con®rms that the uppermost 5 km of the crust must be slower than in model M1, similar to our model MN6, nonetheless better models and their uncertainty can also be found. The warning of the present paper, that the available crustal model is not completely appropriate, is also common to the other methods. Even simple methods (e.g., those inverting only the peak P and S amplitudes by means of a homogeneous half-space), apparently free of the structural uncertainties, are equally vulnerable. They may ®t the peak amplitudes very well in an erroneous model by mapping the structural uncertainties into the focal mechanisms, non-DC components, time function complexities, etc. It is very dicult to compete with methods like that since their mis®t functions re¯ect neither the large unmodeled part of records, nor the inadequacy of the structural model, thus their errors are formally very low. The same applies for methods inverting only selected components. Another question is whether the results could improve when increasing bandwidth towards lower frequencies. Obviously, at larger wavelengths the wave®eld is less sensitive to the unknown crustal structure details. This, however, would require solution of the instrumental event-induced instabilities during nearby M ³ 3 events, which seem to exist not only in CMG-3T, but also in the other BB instruments (C. Guralp and G. Holcomb, pers. comm.). They have not been widely discussed in seismological literature since the BB stations are primarily used for other applications, free of this problem. The studied frequency band should include not only low frequencies, because at long wavelengths we would lose the ability to optimize the depth. Although originally developed for small earthquakes M < 4 and nearby stations, the method is also expected to be useful for larger events observed at a few regional BB stations, but still too weak to allow global study. The larger magnitudes will increase the desired low-frequency content, and the larger epicentral distances will help to avoid the event-induced instrumental instabilities. At regional distances, the role of the polarities will become small (often unclear, or dicult to model by the ray method). However, the advantage of the processing records as a whole will further increase, since Pg, Sg, Lg waves must be modeled as complete (interference) wave®eld, not individual arrivals. Of course, for applications like that the BB stations must have considerably larger interstation separation compared to the three Corinth Gulf stations studied in this paper. The ASPO code written in Fortran77 is available from the ®rst author of this paper.

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Pure appl. geophys.,

Acknowledgments This work would never have been possible without the endless enthusiasm and support of Prof. G.-A. Tselentis (Director of the Seismological Lab., Patras Univ.), and his sta€. J. BrokesÏ ovaÂ, M. KvasnicÏka, I. OprsÏ al and V. Plicka (Charles Univ.) participated in the ®eldwork and the data collection. Two broadband stations were purchased by the Faculty of Mathematics and Physics, Charles University. The third station was ®nanced by the Inco-Copernicus COME project. The contributions from the Inco-Copernicus project, ISMOD, the Charles University grant 5/97/B, MSMTJ13/98: 113200004, GACR 205/00/0902 and NATO EST.CLG.976035 grants are also acknowledged. The paper pro®ted from constructive reviews by J. SÏõ leny (Acad. Sci., Prague) and an anonymous reviewer.

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