Earthquake magnitude estimation from early radiated energy

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Earthquake magnitude estimation from early radiated energy
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  Earthquake magnitude estimation from early radiated energy Gaetano Festa, 1 Aldo Zollo, 1 and Maria Lancieri 2 Received 4 August 2008; revised 14 October 2008; accepted 16 October 2007; published 27 November 2008. [ 1 ] From inspection of a large set of Japanese events, weinvestigate the scaling of the early radiated energy, inferredfrom the squared velocity integral (  IV  2) with the finalmagnitude of the event. We found that the energy can onlydiscriminate whether the event has a magnitude larger or smaller than 5.8, and in the latter case it can allow for real-time magnitude estimation. However, by normalizing  IV  2for the rupture area, the initial slip scales with the magnitude between 4 <  M   < 7 following the expected scaling laws. Weshow that the ratio between the squared peak displacement and  IV  2 is a proxy for the slip following the same scaling but it can be directly derived from the data, without anyassumption on the rupture area. The scaling relationship between initial slip and magnitude can be used for earlywarning applications, when integrated in a probabilistic,evolutionary approach.  Citation:  Festa, G., A. Zollo, and M.Lancieri (2008), Earthquake magnitude estimation from earlyradiated energy,  Geophys. Res. Lett. ,  35 , L22307, doi:10.1029/ 2008GL035576. 1. Introduction [ 2 ] Earthquake early warning systems are real-time mon-itoring infrastructures designed to provide a rapid notifica-tion of the potential effects of an impending earthquake,through the fast telemetry and the processing of data fromdense instrument arrays deployed in the source region or surrounding the target site.[ 3 ] As a first order approximation, the amplitude and thecharacteristic frequency of the seismic records depend onthe event location and magnitude and on the attenuationmechanisms that the waves undergo during the propagationfrom the source to the site of interest. The latter effect isalmost independent of the event size and it can be modeledwith sufficiently high accuracy. On the other hand, theearthquake location can be very rapidly determined fromearly signals recorded at a few stations close to the hypo-center [  Horiuchi et al. , 2005]. Therefore the lead time of anearly warning system critically depends on the systemcapability to predict the final earthquake magnitude frommeasurements on early recorded signals.[ 4 ] Correlations between the final magnitude of an earth-quake and several parameters measured on the early por-tions of P-waves and S-waves have been widelyinvestigated for seismic early warning applications. Theanalyses of the seismic databases from southern California[  Allen and Kanamori , 2003;  Olson and Allen , 2005;  Wu and  Zhao , 2006], Taiwan [ Wu and Kanamori , 2005], the Euro-Mediterranean region [  Zollo et al. , 2006] and Japan [  Zolloet al. , 2007;  Lockman and Allen , 2005] show that the predominant period and the peak of ground displacement scale with the final magnitude over a wide range (4 <  M   <8), arguing that the energy available to break new asperitiesmay already differ at the initial stage of the rupture [ Olsonand Allen , 2005;  Zollo et al. , 2006].[ 5 ] Although the observations indicate that the early peak and the dominant frequency of seismic signals do increasewith the magnitude, the limit at which the prediction isreliable remains debatable.  Rydelek and Horiuchi  [2006]and  Rydelek et al.  [2007] have claimed that neither theinitial predominant period nor the peak ground displace-ment significantly increase with a magnitude beyond  M   = 6.Moreover geological observations support the hypothesisthat the arrest mechanisms of large earthquakes are mainlyassociated with structural features in the fault geometry,irrespective of the event size [ Wesnousky , 2006].[ 6 ] For early warning applications, the ground motionestimations associated with the more destructive S waverely on the information carried out by the first seconds of the P wave at the same recording site. When the target site isfar from the hypocenter (distances larger than 80 km;regional early warning), the S wave recorded close to thefault can help in constraining the ground motion estimateswithout significantly increasing the system lead-time. Nearbythe epicenter, S data can even provide the only estimation of the magnitude, when the S minus P time is smaller than theP window (few seconds) required for the estimation of themagnitude.[ 7 ] In this work we investigate the scaling of the radiatedenergy, inferred from squared velocity integral measured onearly P- and S-waves signals, with the final size of theevent. From the analysis of moderate to large earthquakes,recorded by the Japanese strong motion networks Kik-net and K-net, we discuss the behavior of the ‘‘macroscopic’’slip at the beginning of the rupture in the magnitude range4 <  M   < 7. 2. Data Analysis [ 8 ] Let us consider the integral of the squared velocity(  IV  2) measured in the early portions of P-waves andS-waves  IV  2 c  ¼ Z  t  c þ D t  c t  c v  2 c  t  ðÞ dt  where the subscript   c  refers to the P or S phase,  t  c  is thecorresponding first arrival, and  v  c  is the particle velocitymeasured on the seismograms. Finally  D t  c  is the length of the signal along which the analysis is performed.  IV  2 has GEOPHYSICAL RESEARCH LETTERS, VOL. 35, L22307, doi:10.1029/2008GL035576, 2008 Click Here for Full Article 1 Dipartimento di Scienze Fisiche, Universita` di Napoli ‘‘Federico II’’, Naples, Italy. 2 Osservatorio Vesuviano, Istituto Nazionale di Geofisica e Vulcanologia, Naples, Italy.Copyright 2008 by the American Geophysical Union.0094-8276/08/2008GL035576$05.00 L22307  1 of 6  the advantage that it includes information about the energyradiated by the advancing rupture on the fault plane, andtherefore it provides direct, although partial, insights intothe physics of the fracture [  Boatwright and Fletcher  , 1984].[ 9 ] We have processed about 2500 high-quality strong-motion records from the K-Net (http://www.k-net.bosai.go.jp/k-net/index_en.shtml) and Kik-net (http://www.kik. bosai.go.jp/kik/index_en.shtml) Japanese databases. Therecords correspond to events that occurred from 1996 to2005 and with magnitudes ranging from 4.0 to 7.0. Welimited the analysis to stations close to the fault, for whichthe hypocentral distance was less than 60 km. The P and S phases were manually picked from the records, and then thevelocities were obtained by integration and band-passfiltering in the frequency band of 0.05–10 Hz. The  IV  2values were grouped into magnitude bins of 0.3, with the bin width related to the errors in the magnitude estimates.For the analysis, we selected D t   p  = 4  s  of signal after the first P arrival and  D t   s  = 2  s  beyond the S arrival.[ 10 ] We stress here that 2  s  and 4  s  on the records do not correspond to 2  s  and 4  s  of rupture when the observer is at near source distances. Since the velocity at which a dynamicrupture advances on a fault   c r   is comparable with thevelocities at which the information travels in an elasticmedium ( c  p  or   c  s  for P and S waves respectively), the imagethat an observer has of an advancing rupture is deformedand stretched in the direction of the observer. Moreover,different observers look at the fault from different views,giving an overall image that relates to the envelope of thescanned areas [  Festa and Zollo , 2006]. Since  c  s  <  c  p , thefault region spanned by the P waves is smaller than the oneimaged by the S waves, for the same time window onseismograms after the initial P- and S-arrivals. Since  c  s  isclose to  c r  , the S wave explores shallow regions in theobserver direction, while the P wave images are almost isotropic around the hypocenter. These are the reasons for which we need a larger P time window to capture the sameinformation as is carried by the S wave.[ 11 ] Figure 1 shows one component of the velocity,integrated from the accelerometric records for two eventsof magnitude  M   = 7.1 and  M   = 5.0 respectively, which wererecorded at about 45 km from the hypocenter. It also showsa logarithmic plot of the squared velocity modulus as afunction of time. The energy increases by more than threeorders of magnitude between the two events, with a steep Figure 1.  (top) One horizontal component (EW) of the velocity recorded at about 45 km from the hypocenter, for twoevents of magnitude  M   = 5 and  M   = 7.1 (the key gives the recording station references). (bottom) The squared velocitymodulus as a function of time for the same events as plotted in Figure 1 (top). Between the two events the level of theenergy drops by more than three orders of magnitude, with a steep change of the trend in correspondence of the P and Swave arrivals. L22307  FESTA ET AL.: EARTHQUAKE MAGNITUDE ESTIMATION  L22307 2 of 6  variation in correspondence of the P and S arrivals and analmost constant level for few seconds beyond them.[ 12 ] To compare records from stations located at different distances from the hypocenter, we normalized the measure-ments to the reference distance  R 0  = 10 km, by analyticallyremoving the geometrical spreading term log(  R 2 /   R 02 ). Thefinal value of the integral of squared velocity is thereforereferred to as  IV  2 c 10km . 3. Radiated Energy and Average Slip [ 13 ] Figure 2 shows the plot for the  IV  2  P  10km (squares) and  IV  2 S  10km (circles) as functions of the final magnitude of theevent. To discuss the behavior of these quantities, we alsoadd the velocity integral evaluated for the whole signal S inthe case of large events (diamonds). When accounting for the whole duration, the points are aligned along a straight line, with a slope ( a  = 1.41 ± 0.04) that is compatible withthe expected scaling factor of 1.5 [e.g.,  Scholz  , 2002].Straight lines with this slope fit both the P and S data upto a magnitude  M   = 5.8. Beyond this, the early energyincreases less, or does not increase at all, with respect to thefinal magnitude. A rupture size having a magnitude  M   = 5.8is comparable with the area imaged by the back-propagationof the selected P- and S-windows.[ 14 ] By interpreting the velocity integral representationsin the light of the scaling laws, for a window time length of 4  s  for P-waves and 2  s  for S-waves, we can conclude that  below  M   = 5.8 the apparent duration is smaller than theinvestigation time window and the increase in the emittedradiation is associated with both the increasing fracture areaand the increasing average slip. Beyond a magnitude of 5.8,the data provide a partial image of the advancing rupture,coming from a fault portion which has almost the same area,despite the magnitude. Any increase in the velocity integralhas to be ascribed to the slip.[ 15 ] Hence, the velocity integral can be used only todiscriminate whether an event has a magnitude larger or smaller than  M   = 5.8 and in the latter case to evaluate themagnitude of the event. We compute regressions lawsthrough the first P and S points up to magnitude  M   = 5.8.The resulting curves are log(  IV  2  p 10km ) =   7.7(±0.3) +1.4(±0.1)  M   for P waves and log(  IV  2  s 10km ) =   6.3(±0.4) +1.4(±0.1)  M   for S waves, where the velocity is measured incm/s. We remark that below  M   = 5.8 any deterministicmethod (such as local  Ml   or moment   Mw  magnitude) can beused to estimate the magnitude since the selected timewindows entirely contain the direct waves emitted by thesource. Beyond  M   = 5.8 the prediction becomes ill-posedand if a scaling with the magnitude exists, it has to concernthe slip.[ 16 ] For this, we transform the velocity integral into theradiated energy  E   [  Kanamori et al. , 1993]:  E  c  ¼ 4 p   R 2  F  2 < 2 c r  cIV  2 c where we use the average values  r   = 2.7 g/cm 3 for thedensity,  F   = 2 for the free surface coefficient,  c  s  = 3.3 km/sand  c  p /  c  s  =  ffiffiffi 3 p   .  < c 2 is the ratio between the actual and theaverage squared radiation pattern that we fixed to 1[  Kanamori et al. , 1993]. We also assume the expectedvalue for the ratio  E   s /   E   p  = 16.7 [  Boatwright and Fletcher  , Figure 2.  Scaling of the 4s velocity integral for the P waves (squares) and 2  s  velocity integral for the S waves (circles) as afunction of the magnitude. Superimposed on the picture we also add the velocity integral over the total S phase for largeearthquakes (diamonds). The error-bars represent the standard deviation error around the mean value. The total velocityintegral points are aligned along a straight path with a least-squares slope compatible with standard scaling laws. Straight lines with this slope fit the first P-wave and S-wave early energy points up to  M   = 5.8. L22307  FESTA ET AL.: EARTHQUAKE MAGNITUDE ESTIMATION  L22307 3 of 6  1984] to obtain the total radiated energy  E   from the P and Senergy estimates. The radiated energy is related to the slip h  D i  as:  E  ¼ 12 D s   D h i  A where  D s   is the stress drop. Since the velocity integralcomputed along the complete signal scales with themagnitude following the theoretical scaling law, we seethat the stress drop must be nearly constant in theinvestigated magnitude range. Therefore, we can estimatethe average stress drop from the level of the velocityintegral, obtaining D s   = 6.7 MPa. We note here that under the hypothesis of constant stress drop, the latter onlyinfluences the absolute value of the slip, and does not affect the scaling of the slip with the magnitude.[ 17 ] Finally, to obtain the slip, we have to normalize theradiated energy for the rupture area as seen from the earlyseconds on the records. For small earthquakes (  M   < 5.8), weuse the moment versus area scaling relationship for acircular crack [  Madariaga , 1976]:  M  0  ¼  167 p  3 = 2 D s   A 3 = 2 where the moment   M  0  is known from the velocity integral.Then, beyond magnitude  M   = 5.8, a constant value for therupture area is used, corresponding to the one estimated for an earthquake of   M   = 5.8.[ 18 ] Figure 3 shows the log of the slip, as imaged fromthe first few seconds of P and S wave records on seismo-grams, as a function of the magnitude of the event. We seehere that the slip from both the P-waves and S-waves scaleswith the magnitude from  M   = 4 to  M   = 7, with the slipestimates from the P and S data very consistent each withthe other. Superimposed on these data, there is the plot of the best-fit lines going through the data points (solid line for S waves and dashed line for P waves). The two curves are: log  D h ið Þ  P  ¼ 0 : 30   0 : 05 ð Þ  M    2 : 6   0 : 3 ð Þ log  D h ið Þ S  ¼ 0 : 40   0 : 05 ð Þ  M    3 : 2   0 : 2 ð Þ with the slopes comparable with  a  = 0.5 of the theoreticallyexpected scaling laws [ Scholz  , 2002]. The high correlationof the mean values can also be quantified through the valuesof the correlation coefficients which are  r   p 2 = 0.86 and  r   s 2 =0.92, for P and S wave respectively.[ 19 ] We note that, assuming a constant stress drop, theradiated energy scales as the product   h  D i  A , while the peak displacement   PD  scales as h  D i 1/2 . Hence, if the scaling lawholds also in the early portion of the signal, the ratio  PD 2 /   IV  2 should scale as the slip. The strength of this parameter is that it represents a proxy for the slip directly measurableon seismic signals. Figure 4 shows the ratio for 4  s  of P-waveand 2  s  of S-wave. The two best-fit curves are: log  PD 2 =  IV  2    P  ¼ 0 : 38   0 : 02 ð Þ  M    3 : 28   0 : 13 ð Þ log  PD 2 =  IV  2   S  ¼ 0 : 48   0 : 03 ð Þ  M    3 : 82   0 : 17 ð Þ Again, the two slopes are comparable with those of Figure 3,as well as with the expected theoretical scaling. The valuesof the correlation coefficients  r   p 2 =  r   s 2 = 0.97 arerepresentative of the high correlation of the mean values.[ 20 ] Although the variation of the slip with magnitude isstatistically significant, the data dispersion in Figures 3 and4 indicates that the inferred scaling relationships obtainedcannot be considered as deterministic laws, since the errors Figure 3.  The slip as a function of the magnitude from the P (squares) and S (circles) early records. The dashed and solidlines represent the best-fit curves through the points P and S respectively. The dotted lines represent the prediction boundsfor a new observation with a 95% confidence level in correspondence of the S best-fit curve. L22307  FESTA ET AL.: EARTHQUAKE MAGNITUDE ESTIMATION  L22307 4 of 6  on the slip are large. Hence, a single measure of slip cannot significantly discriminate an event of magnitude  M   = 6 froman event of magnitude  M   7. When drawing on Figure 4 ahorizontal line in correspondence of the value of slipexpected for a magnitude  M   = 6, the probability that avalue smaller than this will belong to the distribution of theslip data for   M   = 7 is 10% and 19% from S and P estimatesrespectively.[ 21 ] However, since the data are normally distributed, theerror on the mean value decreases when extracting infor-mation from several receivers which are representative of the slip distribution corresponding to a given value of magnitude. In such a case, the uncertainty associated withthe mean becomes comparable with the error from the best-fit curve, which is about 0.5 in magnitude. 4. Conclusions [ 22 ] The squared velocity integral measured on the first few seconds of P and S waves as a function of the finalmagnitude of the event shows a two-slope curve withseparation limit at about   M   = 5.8. For   M   < 5.8, both 4  s  of P-wave and 2  s  of S-wave image the whole rupture process,and in this range  IV  2 can be used as an estimator of themagnitude of the event. Beyond  M   = 5.8 the area imaged bythe first few P and S records remains almost the samedespite the magnitude and any increase in  IV  2 vs  M   plotshas to be ascribed to the variation of the slip. Under thehypothesis of constant stress drop, the initial slip can beachieved by normalizing the radiated energy by the rupturearea as seen by the first P- and S-wave segments. Therupture size up to magnitude  M   = 5.8 was estimated usingthe moment versus area relationship for circular crack whilea constant value, corresponding to  M   = 5.8, was used for large earthquakes. After normalizing by the rupture area, alinear relationship between the slip and the magnitudeyields in the whole magnitude range, whose slope iscomparable with that from standard scaling laws. This result implies that, for 5.8 <  M   < 7, the average slip measured inthe early stage of the rupture looks similar to the finaldistribution of the slip on the fault plane, according to thehypothesis that large slip zones are likely to be locatedwithin or close to the hypocenter [  Mai et al. , 2005].Additionally, the slip in the initial part of the rupture hasreached or is closed to its final value, indicating that the risetime should be associated with a time scale smaller than 4  s of the P-wave and 2  s  of the S-wave.[ 23 ] Under the same assumptions, the trend of the slipwith magnitude can be directly inferred by measurementson the data, through the ratio  PD 2 /   IV  2. We prove that thisquantity follows the same scaling as the theoretical model,with smaller errors in the estimations of the best-fit coef-ficients. Although the errors on the single measurement of   PD 2 /   IV  2 are too large for a deterministic prediction of themagnitude, several measurements can be combined in a probabilistic evolutionary framework for early warningapplications. If   PD 2 /   IV  2 values are representative of the probability function at the corresponding magnitude, theerror in the final estimation can be pushed down to about 0.5 units.[ 24 ]  Acknowledgments.  Authors acknowledge AMRA Scarl, ‘‘Ana-lisi e Monitoraggio del Rischio Ambientale’’ for financial support and twoanonymous reviewers for their valuable contribution in improving themanuscript content. References Allen, R. M., and H. Kanamori (2003), The potential for earthquake earlywarning in southern California,  Science ,  300 , 5620, 786–789,doi:10.1126/science.1080912. Figure 4.  Scaling of the ratio  PD 2 /   IV  2 as a function of the magnitude in the early portion of the P-waves and S-waves.The scaling is comparable to that obtained in Figure 3. The dotted lines represent the prediction bounds for a newobservation with a 95% confidence level in correspondence of the S best-fit curve. L22307  FESTA ET AL.: EARTHQUAKE MAGNITUDE ESTIMATION  L22307 5 of 6
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