Magnitude-Recurrence Relationship and its Effect on Uniform Hazard Spectra: A Current Assessment

Transcription

1 Magnitde-Recrrence Relationship and its Effect on Uniform Hazard Spectra: A Crrent Assessment T. K. Sen PhD (London), CEng., FICE Granherne, UK, a ssidiary of Kellogg Brown & Root Ltd SUMMARY: Gtenerg and Richter showed that magnitde-recrrence relationship may e represented y a linear relationship when the log of annal rate of exceedance was plotted against magnitde. This type of recrrence model has een in se ecase of its simplicity and ecase it fits the data reasonaly well over a sefl range of magnitdes of engineering interest. A qadratic log freqency relationship etween magnitde and the mean annal rate has een proposed y others and shown to fit the availale data well in the high magnitde range of the magnitde-recrrence relationship. UHS (Uniform Hazard Spectra) sing a qadratic log freqency and the linear relationship, has een developed and compared. It is shown that sing a linear relationship wold reslt in a more conservative design when defining UHS for dctility level earthqakes (DLE) (retrn periods > 5, years). However, UHS for strength level earthqakes (SLE) (retrn periods < years) are not affected greatly y the se of a linear relationship. Keywords: U.H.S., magnitde, recrrence, qadratic, relationship. INTRODUCTION How often do earthqakes occr? Does some kind of pattern exist in the time scale at which earthqakes are known to occr? In a seismologically active region earthqakes occr at irreglar intervals of time. It is fairly ovios that in order to extract a meaningfl pattern, the length of the record mst e reasonaly large. The longer the record the etter it is. Historical records are dated and hence it is possile for the seismologists to analyse and assess the recrrence relationship. Gtenerg and Richter (954) introdced the magnitde-recrrence relationship gathering data from earthqakes in the sothern California region. The data (spread over a certain length of time) was organised in a manner to reflect the nmer of earthqakes that exceeded a certain magnitde. Also from the organised data, the mean annal rate of eing exceeded, M, of an earthqake magnitde, M, was defined as the occrrences greater than M divided y the length of the time period. Or in other words the average rate at which an earthqake of some size will e exceeded. The reciprocal of the mean annal rate of eing exceeded was referred to as the mean retrn period of the earthqake. It was fond that the logarithm of the annal rate of exceedance of sothern California earthqakes, plotted against earthqake magnitde, reslted in a linear relationship. They proposed a relationship of the form: log m a m. Where a and are field constants for a particlar region. The validity of Gtenerg-Richter relationship has een qestioned y varios investigators (Schwartz and Coppersmith, 984, Yongs and Coppersmith, 985). Becase of its simplicity and as it fits the oserved earthqake data reasonaly well over a sefl range of engineering interest, this model has een fond to e convenient and is in general se today.

2 . Backgrond Use of Qadratic Form: One of the concerns regarding the se of a linear relationship, if n-trncated, is that it normally overestimates the occrrence of large events and ecase of the scattering of data associated at the pper ond range, it does not reflect the tre state. Shlien and Toksoz (97) have shown that, at least for magnitdes for which reliale data are availale (USCGS, Dda, 965), a qadratic log freqency relationship fits well. Merz and Cornell (973) introdced a more general case of a qadratic magnitde-freqency relationship with a finite pper-ond, m. ' ' m 2 log a ( m m ) ( m m ) (.) And the cmlative distrition fnction as: * 2 2 m 2 F ( m) P M m k [ exp{ ( m m ) ( m m )}] (.2) M [Both M and m are sed as symols for magnitdes. In general, the notation adopted is that the pper case is sed when variale referred to is a random variale; the lower case is sed when referring to oserved vales of that variale]. Where k exp{ ( m m ) ( m m )} * 2 2 m 2 ' ' Otline of the Paper. The qadratic form for magnitde recrrence relationship is investigated next. The qadratic form adopted is compared with the dataase of USCGS cataloge and those otained y Dda and Gtenerg (964) over a period from (see Fig. ). 2. On the asis of the qadratic relationship adopted (step ) niform hazard spectra for retrn periods for SLE (5 and years) and DLE (, years) are derived next. 3. UHS ased on a linear relationship which is a close approximation of the dataset, except in the high magnitde range (see Fig. ) are derived next. 4. The nmerical engine for developing the UHS is the Monte Carlo simlation techniqe (Sen, 26, 29) and is riefly otlined here. 5. In the conclding part, the effects of the qadratic form on the UHS, which is the main ojective of this paper, are discssed and conclsions drawn.

3 log (earqakes/year) 2. QUADRATIC MAGNITUDE-RECURRENCE RELATIONSHIP Referring to eqation (.), the more general case of a qadratic magnitde-freqency relationship with a finite lower ond mand an pper-ond m (proposed y Merz and Cornell, 973) is sed here and expressed as: log ( ).298( ) 2 m m m m m (.3) The qadratic form adopted is compared with the dataase of USCGS cataloge and those otained y Dda and Gtenerg (964) over a period from and is shown in Fig.. [The constants have een modified to sit the USCGS and Dda (965) data as presented y Shlien and Toksoz (97)] scgs data Qad Mag linear rel. mag (M) log m ( m m ) log ( m m ).298( m m ) m Figre. Plot of USCGS Data Points and the Qadratic Recrrence Relationship Adopted 3. CUMULATIVE DISTRIBUTION FUNCTION (CDF) CDF otained from eqation (.2) with m =4 and eing exceeded). m = 8 is shown in Fig.2 (Q is the proaility of M.8 Q Samples Cmlative Distrition Fnction for Qad-Mag Rec rrence Relation Figre 2. Cmlative Distrition Fnction for Qad-Mag Recrrence Relation

4 3.. General Scheme Monte Carlo Simlation Monte Carlo simlation is an estalished techniqe to solve proailistic models. The proailistic model otlined y Cornell (968) may e solved y the Monte Carlo simlation process. The nmerical approach is different. It mirrors real life events as may e evinced from the comptational scheme shown in the flow chart (see Fig.3). Sample proaility distritions to otain a point estimate for each independent inpt variale Calclate dependent variales Calclate otpt variales Post-process reslts to otain statistical description of otpt variale Iterate Figre 3. General Scheme of Monte Carlo Simlation All comptations for generating plots of exceedance of acceleration against retrn periods, were carried ot with the mathematical software Mathcad 4. The software has a random nmer generator incorporated within which is sed to otain a point estimate for each independent inpt variale. 3.2 Preliminary Steps The preliminary steps are as follows:. Falt Line As an example the following were considered: The falt line is 65 km long and earthqakes can occr anywhere along the falt line. Ths, niform random distrition may e assmed. The following nmerical reslts are otained for a site located a minimm srface distance,, of 4 km from a line sorce of earthqakes at a depth of 2 km. In the aove intensity relationship R is related to the position where the earthqake originates which happens to e randomly distrited. 2. Relationship Used Before we consider constrcting the niform hazard spectra for a site, we mst know the proaility of peak grond motion (peak acceleration, velocity etc) eing exceeded. Cornell s proailistic methodology otlined aove can e applied on any fnctional relationship etween site grond motion variale Y (peak acceleration, velocity and displacement), and M and R. Thogh not qite appropriate this wold nevertheless e adeqate for illstrating the effects of the qadratic form on the UHS, which is the main ojective of this paper. The fnctional relationship provided y McGire (974) working on data from sites in Western USA, is shown elow: or S ' 2 M 3 ( R 25) cm/sec 2 (.4) a log S a M log( R ) (.5)

5 Retrn Period (Years) The constants for the aove eqation are shown in Tale 3. S ' a S a 2 M 3 ( R 25) log M log( R ) (with permission, reprodced from Dowrick, 987) Tale 3. McGire s Attenation Expressions for Spectral Acceleration with 5% Damping (974) Period (s) S a ' 2 3 Coeff. of Var. of Monte Carlo Simlation Plots for Peak Grond Acceleration We can apply the Monte Carlo process to derive the plot for peak grond acceleration vs. retrn period, R, (or proaility of eing exceeded). The grond acceleration exceedance crve for period =.2 sec is shown in Fig F i N A i Acceleration (cm/sec^2) - Period.2 sec Plot of Gr Acc vs Retrn Period Figre 4. Grond Acceleration Exceedance Crve for Period =.2 sec

6 Grond Acc (cm/sec^2) Retrn Period Spectral Acceleration/Velocity 3.4 Uniform Hazard Response Spectrm (UHRS) 3.4. Qadratic-Magnitde Recrrence Relationship Step Calclation of peak grond accelerations for periods. 4. sec (as shown in Tale 3) is carried ot first (plots not shown here) y modifying the Mathcad, V4 sheets. Step 2 The concept and methodology is shown in Fig. 5. a c d a c d Spectral Acceleration/Velocity a c d Period Figre 5. Constrction of UHRS Step 3 Finally, we follow the plan otlined in Fig. 5 for constrcting the UHRS. The plot for UHRS for a retrn period of 5, and, years is shown in Fig UHRS - Linear Relationship The linear relationship adopted for comparison with the qadratic relationship is shown in Fig.. The relationship is of the form: log m ( m m ) Years years 4 2,5,5 2 2,5 3 3,5 4 4,5 Period (secs) Figre 6. Plot of UHRS (Retrn Period: 5, and, years)

7 grond acceleration (cm/sec^2) grond acceleration (cm/sec^2) grond acceleration (cm/sec^2) 4. CONCLUDING REMARKS Comparison of the UHS developed with the qadratic and the linear relationship is shown in Figs. 7, 8 and 9. 5 year retrn event Period (secs) Qadratic Relationship Linear Relationship Figre 7. 5 year Retrn Event, year retrn event Period (secs) Qadratic Relationship Linear Relationship Figre 8., year Retrn Event, year retrn event Period (secs) Qadratic Relationship Linear Relationship Figre 9., year Retrn Event The qadratic relationship predicts fewer events of high magnitdes and as expected, Fig. 9 shows the difference in sing a qadratic instead of a linear magnitde-freqency relationship and is significant for the, year retrn event, as grond accelerations for this low-risk event are higher.

8 In the UHRS for the 5 and, year retrn period, where grond accelerations expected are mch lower, and there is no appreciale difference in UHS from an engineering perspective (see Figs. 7 and 8). Merz and Cornell (973) had arrived at a similar conclsion when risks associated with high grond accelerations were compared with the two magnitde-freqency recrrence relationship. ACKNOWLEDGEMENT The athor wold like to thank the Granherne and KBR management for permission to present this paper. REFERENCES Cornell, C.A. (968), Engineering Seismic Risk Analysis, Bll. Seismol. Soc. Amer., 58, Dowrick, D.J., (987), Earthqake Resistant Design, John Wiley & Sons, Chichester. Dda, S.J. (965), Seclar Seismic energy Release in the Circm-Pacific elt, Tectonophysics, 2(5), Gtenerg, B. and Richter, C.F. (954), Seismicity of the Earth, Princeton University Press. McGire, R.K. (974), Seismic strctral response risk analysis incorporating peak response regressions on earthqake magnitde and distance, Res. Report R74-5, Dept of Civil Engineering, Massachsetts Institte of Technology. Merz, H.A. and Cornell, C.A. (973), Seismic Risk Analysis Based on a Qadratic Magnitde-Freqency Law, Bll. Seismol. Soc. Amer., Vol 63, No. 6, Schwartz, D. P. and Coppersmith, K. J. (984), Falt ehavior and characteristic earthqakes:examples from the Wastch and San Andreas falt zones J. Geophysical Research, Vol 89, No B7, Sen, T.K., (26), Constrction of Uniform Hazard Response Spectra Using Monte Carlo Simlation, Proc. First Eropean Conf. Earthqake Engineering and Seismology, Geneva, Switzerland, 3-8 Sept. 26. Sen, T.K., (29), Fndamentals of Seismic Loading on Strctres, John Wiley & Sons, Chichester, UK. Shlien, S. and Toksoz, M.N. (97), Freqency-Magnitde Statistics of Earthqake Occrrences, Earthqake Notes, Vol. XLI, No.. Yongs, R. R. and Coppersmith, K. J. (985), Implications of falt slip rates and earthqake recrrence models to proailistic seismic hazard estimates. Bll. Seismol. Soc. Amer., Vol. 75,