International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Sensitivity of ground motion simulation validation criteria to filtering Khoshnevis, Naeem; Taborda, Ricardo Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Sensitivity of Ground Motion Simulation Validation Criteria toFilteringNaeem KhoshnevisGraduate Student, Center for Earthquake Research and Information, University ofMemphis, Memphis, USARicardo TabordaAssistant Professor, Department of Civil Engineering, and Center for EarthquakeResearch and Information, University of Memphis, Memphis, USAABSTRACT: The validation of ground motion synthetics has received increased attention over the lastfew years due to advances in physics-based deterministic and hybrid simulation methods. Validationof synthetics is necessary in order to determine whether the available simulation methods are capableof faithfully reproducing the characteristics of ground motions from past earthquakes. Some validationmethods use filters to evaluate the quality of the fit between synthetics and data within different frequencybands. This is done, primarily, to weight the contribution of different wavelengths so that low frequenciesare given more weight than high frequencies. One particular method of interest is the goodness-of-fit(GOF) criterion introduced by Anderson (2004). In this method, the degree of similarity between twosignals is quantified by means of a set of ten (error) metrics which are projected on a 0–10 scoring scale.These metrics are based on ground motion characteristics commonly used in seismology and earthquakeengineering. The scores are used to evaluate each given pair of signals in the three components of motionand within different frequency ranges. The scores of each frequency band, component, and metric arethen combined to obtain a final GOF score. In this paper we study the sensitivity of the GOF scores,and thus the sensitivity of the validation itself, to the filtering process when different filter parametersare considered. Our initial analysis of results show that GOF methods are susceptible to the design offilters. The filter’s order, for instance, seems to significantly affect the interpretation of the validation—especially for metrics that are time-dependent (e.g., peak ground response). We evaluate the implicationsof the variability in GOF scores on the case study of the 2008 Mw 5.4 Chino Hills earthquake, andinvestigate the sensitivity of GOF criteria to the type of filters used to decompose the signals. We analyzethe consistency and correlation of the results obtained using various metrics by means of a filteringfitting function. Our work indicates that elliptic infinite impulse response filters lead to more reliableresults, over other more commonly used filters; and we provide guidance on the selection of elliptic filterparameters.1. INTRODUCTION AND BACKGROUNDWith recent advances in earthquake ground motionsimulation using physics-based methods and hybridapproaches (e.g., Bielak et al., 2010; Graves andPitarka, 2010), verification and validation (V&V)of synthetics have become increasingly relevant.V&V imply processing signals in preparation toperform qualitative and quantitative comparisons.We are particularly interested in investigating theeffect that different filtering procedures and param-eters can have on quantitative validations done us-ing the goodness-of-fit (GOF) method introducedby Anderson (2004). In this method, the degreeof similarity between two signals is quantified bymeans of a set of ten (error) metrics, labeled C1through C10. These metrics include ground mo-112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015010203040500 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10CountScore Score ScoreN = 60 N = 180 N = 360Figure 1: Variation of PGA GOF scores for the case study of the 2008 Mw 5.4 Chino Hills, California, earth-quake using FIR Chebyshev filters with order N = 60 (left), 180 (center) and 360 (right). Top frames show con-tours drawn based on GOF values derived from 336 stations with records used for the comparisons against syn-thetics from Taborda and Bielak (2013). Bottom frames show histogram distributions in the GOF scale introducedby Anderson (2004).tion characterization parameters commonly used inseismology and earthquake engineering. The quan-titative comparisons are projected on a 0–10 scor-ing scale, where a score of 10 corresponds to a per-fect match. When using Anderson’s method, signalpairs are compared by component of motion and atdifferent frequency ranges to weight the contribu-tion of different wavelengths so that low frequen-cies are given more weight than high frequencies.The scores of each frequency band, component, andmetric are then combined to obtain a final GOFvalue.In two initial studies (Khoshnevis and Taborda,2014a,b) we found that filtering, in particular, canplay a significant role in the output of the GOFscores. In Khoshnevis and Taborda (2014a) weshowed that the variation of the order of a finite im-pulse response (FIR) filter has a substantial influ-ence on the GOF scores, especially on those met-rics that are strongly time dependent. Therefore,among the metrics considered in the GOF method,the Arias intensity score, and the peak acceleration,velocity and displacement scores are most sensi-tive; followed by the scores derived from the energyintegral, energy duration, Arias duration, cross cor-relation, response spectra and Fourier spectra; withthe latter being the least sensitive of all.The overall effect of different filter orders on thevalidation of simulations is illustrated in Figure 1.This figure shows the contour maps of the GOFscores obtained for a simulation of the 2008 Mw5.4 Chino Hills, California, earthquake (Tabordaand Bielak, 2013). These maps are composed us-ing the GOF values obtained from comparisons be-tween 336 signal pairs (synthetics and data). In par-ticular, the maps correspond to the GOF scores forpeak ground acceleration (PGA) metric C5 in An-derson (2004), where Chebysehv FIR filters withorder N = 60, 180 and 360, were used in the band-pass frequency analysis. We recognize these levelsof a filter’s order may sound very high in the contextof seismology and earthquake engineering, whereother types of filters have been traditionally pre-ferred, yet as we will see in the following section,they are perfectly normal. In all, Figure 1 showsthat GOF maps can be influenced by the filter char-acteristics.More important, even though FIR filters ap-proach a perfect filter with increasing order, wehave found that GOF scores do not necessarily con-212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20151.61.41.21.00.80.60.4 0 200 400 600 800 1000OrderNormalized ScoreFigure 2: Variation of PGA GOF scores with increas-ing order of FIR Chebyshev filter for the same 336 sta-tions used for the validation of the Chino Hills earth-quake simulation.verge with increasing filter order. Figure 2 showsthe variation of PGA GOF scores with filter orderfor the same 336 signal pairs from the Chino Hillssimulation. Each score trend is normalized with re-spect to the value obtained with the highest order.Despite the chosen normalization, the substantialvariability over the range of order values suggeststhat, even for higher order filters, there is not onestable score.Using different types of filters can also be influ-ential on the results. In Khoshnevis and Taborda(2014b) we implemented a Chebyhsev I infinite im-pulse response (IIR) filter, instead. As we will ex-plain in greater detail below, IIR filters have sharpercutoffs at the corner frequencies, yet they presentripples in the passband. Figure 3 shows the fre-quency response of a FIR and an IIR bandpass filterand a comparison of a portion of the Fourier ampli-tude spectra around the cutoff frequency of a signalfiltered using these two type of filters, along withthe unfiltered signal. As it can be seen in this fig-ure, the amplitudes of the FIR filtered signal andthe original signal are nearly identical within a por-tion of the passband, but they differ from each otherover a wider transition zone than that of the IIR fil-ter. In turn, the IIR filter does not have a perfectmatch within the passband, but it exhibits a sharperdecay past the cutoff frequency.Arguably, for most applications, these differ-ences are minor and inconsequential. Their ef-0 0.2 0.4 0.6 0.8 1−2000−1500−1000−5000500–20–15–10–5050 0.1 0.2 0.3 0.40204060803.1 3.2 3.3 3.4 3.5 3.6Normalized FrequencyMagnitude (dB)Frequency (Hz)AmplitudeIIRFIROriginalCutoff Freq.Figure 3: Comparison between FIR and IIR filters.Top: amplitude response (dB) of the bandpass filters’design in normalized frequency for corner frequenciesequal to 0.1 and 0.3 of the maximum frequency. Bot-tom: detail of a Fourier amplitude spectra around theupper cutoff frequency of a signal filtered with both IIRand FIR filters (the original signal is also included).fect on GOF scores used for validation may, how-ever, become relevant, especially in the absence ofa well-accepted standard. Moreover, since simu-lations involve numerous unknowns and assump-tions, it is imperative to minimize the additionaluncertainties that post-processing of the data mayhave on the analysis of results. In this paper weseek to contribute to reducing such additional un-certainties by investigating the characteristics of fil-ters used in ground motion signal processing forvalidation purposes.While it would be difficult to decide on a finalideal filter, it is desirable to define a filter that can beused consistently in validation studies with minoreffects on the analysis. Here, we contribute to such312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20151 + δ11 – δ1δ2|H(ejω)|ωp ωs πStopbandTransitionPassbandRippleAttenuationFigure 4: Specification for effective frequency responsefor the case of low-pass filter. (the discrete-time sys-tem).an objective. We review the characteristics of themost commonly used FIR and IIR filters, and makeinitial recommendations on the selection of suitablefilter for ground motion simulation validation.2. FINITE AND INFINITE IMPULSE RE-SPONSE FILTERSFilters are a particularly important class of lineartime-invariant systems. Strictly speaking, the termfrequency-selective filter suggests the existence ofa system that passes certain frequency componentsand rejects all others. In practice, however, filtersare far from ideal, and there are different kinds offilters, each of them with advantages and disadvan-tages. To aid the discussion, in Figure 4, we showthe main characteristics of a low-pass filter in thefrequency domain. Here, δ1 is the maximum rip-ple’s amplitude in the passband, δ2 is maximumripple’s amplitude in the stopband, ωp is the nor-malized cut-off frequency in the passband, and ωsis the normalized cut-off frequency in the stopband.Digital filters can be broadly classified into twogroups: infinite impulse response (IIR) and finiteimpulse response (FIR). FIR filters are very attrac-tive because they are inherently stable (do not havepoles) and can be designed to have linear phase. InFIR filters, the higher the order, the closer to anideal filter. FIR filters, however, tend to have widertransient zones (see Figure 4) and become compu-tationally expensive with increasing orders, due tothe underlying convolution computation. FIR filtersare almost entirely restricted to discrete-time im-plementations. Consequently, their design is basedon directly approximating the desired frequency re-sponse of the discrete-time system. The simplestmethod of FIR filter design is called the windowmethod. This method generally begins with an idealdesired frequency response that can be representedasHd(e jω)=∞∑n=−∞hd[n]e− jωn , (1)where hd[n] is the corresponding impulse responsesequence. Many idealized systems are definedby piecewise-constant or piecewise-functional fre-quency responses with discontinuities at the bound-aries between bands. As a result, these systemshave impulse responses that are non casual and in-finitely long. The most straightforward approach toobtain a causal FIR approximation to such systemsis to truncate the ideal response. This, however, in-troduces undesired wiggles. This is known as theGibbs phenomenon (Oppenheim et al., 1989). Dif-ferent windowing functions have been defined to re-duce this effect. For our study, we examined fourcommonly used windowing methods: Chebyshev,Hamming, Hanning, and Gaussian.IIR filters, on the other hand, are attractive be-cause they are continuous in time, and thus canbe used in real-time applications. They have alsoenjoyed preference because they transitioned natu-rally from analog to digital technologies, thus theycontinued to be used in practice as the technol-ogy evolved over the years. Typical frequency-selective continuous-time approximations are: But-terworth, Chebyshev, and elliptic filters. The for-mer two being traditionally preferred in earthquakeengineering for no evident reason other than, per-haps, the fact that closed-form design formulasof these continuous-time approximations facilitatestheir design. A Butterworth continues-time filter,for instance, is monotonic in both the pass-bandand the stopband. A Type I Chebyshev filter has anequiripple characteristic in the passband and variesmonotonically in the stopband. An elliptic filter isequiripple in both the passband and the stopband.All these approximation methods yield digital fil-412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015FilteredIdeally Filt.Avg. Amp.ScoreCutoff Freq.0204060801000 1 2 3 4 02468101 HzFrequency (Hz)Amplitude ScoreStopband PassbandFigure 5: Evaluation of coherency between the idealfrequency response and filtered signal’s response. Leftvertical axes is the amplitude and right vertical axes isthe score. The mean of amplitude in the inside sectionis shown as dashed line.ters with non-constant group delay or, equivalently,non-linear phase. The greatest deviation from theconstant group delay occurs, in all, cases at the edgeof the passband or in the transition band. If phaselinearity is not an issue, then elliptic approximationyields the lowest order system function, and there-fore elliptic filters will generally require the leastcomputation to implement a given filter specifica-tion (Oppenheim et al., 1989).Altogether, the primary advantage of IIR filtersover FIR filters is that the former typically meeta given set of specifications with a much lowerfilter order than its corresponding FIR filter. Al-though IIR filters have nonlinear phase, data pro-cessing using software such as MATLAB R© is com-monly performed at a post-processing stage (off-line), and therefore the entire data sequence is avail-able for filtering. This allows one to use non-causal,zero-phase filtering approach by forward and back-ward filtering (two-pass) the signals (e.g., via thefiltfilt function). This process eliminates thenonlinear phase distortion of IIR filters. Here, weadopt this strategy and all of our filters are zerophase. Consequently, we do not address phase vari-ations and concentrate only on amplitude effects.3. EVALUATION METHODIt follows from the previous section that designinga filter entails a trade-off between the accuracy ofthe response of the passband, the transition zone,and the stopband. Different applications will there-fore require different methods to assess the efficacyof a filter. The most common alternative for thisis to compare the Fourier amplitude spectra. Thisevaluation is typically done visually. We, however,seek to define standards that provide a solid refer-ence framework for simulation validation. That is,we are interested in filters that are good for narrowbands at low frequencies and wider bands at higherfrequencies, with sharp transition zones, minimumripples in the passband amplitudes, and with suffi-cient attenuation in the stopbands (see Figure 4).In the context of the GOF analysis, Anderson(2004) defined the expression to evaluate the sim-ilarity between two Fourier amplitude spectra as:S (p1, p2) = 10exp{−[p1 − p2min(p1, p2)]2}, (2)where p1 and p2 are the frequency amplitude oftwo waveforms evaluated at every frequency. Sincethe Fourier amplitude score in Anderson’s GOFmethod is the least sensitive parameter to filtering,we are interested in using a similar function to eval-uate the filters’ selection.Eq. (2) decreases monotonically as the differencebetween the parameters p1 and p2 increases. Whilethis offers a good measure for almost all scales, itcannot characterize the differences when one of theparameters vanishes (becomes zero). This is im-portant because in order to quantify the efficacyof the filtering process, we also need to considerthe values in the stop band, which should be com-pared to zero. Therefore, inspired in (2), we definea new scoring function specifically to evaluate fil-ters based on the Fourier amplitude in the stopband.The proposed expression is:S (p1, p2) = 10exp{−[p1 − p2F(a)]2}, (3)where F(a) is a function of the amplitude insidethe passband. The scoring scale defined by Eq. (3)512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015HammingGaussianChebyshevHann–300–250–200–1500500 0.2 0.4 0.6 1.00.8–50–100Normalized FrequencyMagnitude (dB)Figure 6: Comparison of the frequency response of fourFIR filters.avoids the division by zero and yet offers a signal-specific measure of the accuracy of the filter withrespect to an ideal filter with amplitude zero in thestopbands.We use Eq. (2) and Eq. (3) to evaluate the efficacyof filters by comparing the Fourier amplitudes offiltered signals with respect to the result that wouldbe expected from an ideal filter, where the ampli-tude in the passband is the same amplitude of theoriginal signal within the cutoff frequencies, andzero elsewhere. Figure 5 shows an example of thiscomparison and the resulting scores, using Eq. (2)for scoring the amplitude similarity in the passbandand Eq. (3) for scoring the filtered signal remnantamplitude in the stopbands. The whole domain forcomparing is max±1 Hz above and below the cut-off frequencies. For the the reference amplitude inEq. 3, we assumeF(a) = 0.25mean([ f1, f2]) . (4)The values are computed for each discrete fre-quency and the final score is the mean of the scoreat all frequencies within the comparison range.4. RESULTSAs we mentioned before, we are interested in defin-ing a standard set of filter parameters that can beused for accurate and consistent GOF analysis invalidation of ground motion simulation. To facili-tate this, and for practical reasons, all the filters weconsidered here are designed using functions in theSignal Processing Toolbox of MATLAB R©.EllipticButterworthChebyshev–300–250–200–1500500 0.2 0.4 0.6 1.00.8–50–100Normalized FrequencyMagnitude (dB)Figure 7: Comparison of the frequency response ofthree IIR filters.We first consider four different FIR filters: Ham-ming, Gaussian, Chebyshev, and Hanning. Theirmagnitude in frequency is shown in Figure 6. Allthese filters were designed using windowing meth-ods with order equal 4000. Note again that FIR fil-ters allow very high orders. Such orders are uncom-mon in signal processing in earthquake engineeringor seismology, where low-order (yet unstable) IIRfilters such as Butterworth have been traditionallypreferred. In Khoshnevis and Taborda (2014a) weshowed that, for FIR filters, GOF values convergeas the filter’s order increases. Figure 6 also showsthat in the case of FIR filters, all four options havevery similar steepness at the cutoff frequencies, yetdifferent levels of attenuation. From these filters,we choose Chebyshev as representative of the FIRfamily.Figure 7, on the other hand, shows the magni-tude of three IIR filters: Elliptic, Butterworth, andChebyshev. These filters were designed with ordervalues of 57, 57 and 17, respectively. We chose adifferent order for latter because higher orders forChebyshev rapidly develop strong ripples. In otherwords, order 17 for Chebyshev is as high an orderas 57 is for Elliptic and Butterworth. In this case,it is easily seen that while both the Butterworth andChebyshev filters have a strong attenuation in thestopband, their decay is not as sharp as that of theelliptic filter. We therefore choose the elliptic fil-ter from the IIR family, as this is a desired charac-teristic. In addition, elliptic filters offer more con-612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Elliptic (IIR)Chebyshev (FIR)–300–250–200–1500500 0.2 0.4 0.6 1.00.8–50–100Normalized FrequencyMagnitude (dB)Figure 8: Comparison of frequency response of FIRand IIR filterstrol over the magnitudes of the passband ripples, thestopband attenuation, and the transition width.Figure 8 shows a comparison between the se-lected FIR and IIR filters, and illustrates the advan-tages offered by the IIR elliptic filters over the gen-eral shape of FIR filters, strong attenuation with asharp transition zone. Choosing the best filter strat-egy and parameters is, however, non trivial. Forvalidation using the GOF method described before,we focus our attention on bandpass filters. Band-pass filtered signals, however, can also be obtainedby low- and high-pass filtering. Thus one needs tothese alternatives as well.To evaluate our selection of elliptic IIR filters,we processed the same set of synthetics and datafor the 2008 Chino Hills simulation used before.We followed different procedures to obtain band-pass filtered signals and evaluated the filtered setsusing Eqs. (2) and (3); and found that low- and thenhigh-pass filtering the signals leads to a less sen-sitive outcome for GOF validation purposes. Thisis due to the fact that by separating the bandpassfiltering process, one has more control over the fil-tering parameters for the two (low and high) cut-off frequencies. There are, however, some trade-offs. Increasing the filter’s order and attenuationfactor of the stopband, and decreasing the maxi-mum amplitude of the ripples in the pass band al-together at once—assuming that each, separately,should bring the filter closer to an ideal condition—does not work. Therefore, a balance must be found.To find this point of balance we did a sensitivityanalysis based on the different options of the ellip-tic filter. In this analysis we assumed that the pa-rameters of low- and high-pass filters are the same,and focused simply on finding the best parametersfor the process. Figure 9 shows the sensitivity ofthe GOF scores with respect to the filtering orderand attenuation factor (Rs) for the case of an ellipticIIR zero phase filter. We can see here that for lowattenuation factors (Rs = 40), GOF converge withincreasing orders at about 10–15. Larger attenu-ation factors, however, may have negative effectsand may even become unstable (e.g., Rs = 150) athigher order values. Similarly, Figure 10 shows thevariation of the GOF scores goodness with vary-ing attenuation factor. Again, increasing Rs gener-RS = 40RS = 100RS = 150024100 20 30 40 60508610 OrderGOF ScoreFigure 9: Variation of GOF scores with increasing thefilter order, and variable attenuation factor (Rs in dB).Order = 9Order = 17Order = 35024100 100 150 200 3502508650 300Filter Attenuation Factor, RS (dB)GOF ScoreFigure 10: Variation of GOF scores with increasing theattenuation factor (Rs), and variable filter order.712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015ally affects the stability of the scores. These resultswere obtained for a subset of 40 signal pairs fromthe Chino Hills test case, considering five differentbands in the ranges 0.1–0.25, 0.25–0.5, 0.5–1, 1–2, 2–4 Hz. In all cases, the amplitude of ripplesin the passband (Rp) was set to 0.001 dB. We foundthat all bands exhibited the same behavior, althoughwith different initial and mean values.5. FINAL REMARKS AND RECOMMEN-DATIONSWe present a study about the effects that differentfiltering approaches have on validation of groundmotion simulations. According to our evaluationof different filters, elliptic IIR filters offer the mostconsistent results for validation purposes, wherenear-ideal filters are desirable. We reached this con-clusion based on a quantitative evaluation of thesimilarity of a filter’s result with that expected froman ideal filter. The quantitative measure introducedhere is based on similar ideas first put forward byAnderson (2004), but modified to facilitate the eval-uation of the stopband alongside the passband. Theproposed scoring method helps find optimal param-eters and serves as a starting point to find the bestpossible filter.Even though we can increase the attenuation ofstopband and decrease the allowable ripple magni-tude in the passband, we found that there is a tradeoff between these factors and modifying them si-multaneously may not always yield better results.After conducting a sensitivity analysis, we foundthat in the case of elliptic filters, an attenuation fac-tor between 35–100, allowable ripple magnitudesin passband less 0.01, and filter orders bigger than17, are good initial values for filters’ design. Whilethe present study focuses on the application of thesefilters for ground motion validation purposes, it ispossible to think that our evaluation may be used inother contexts, especially in seismology and earth-quake engineering, where other filters have beentraditionally used without much attention.We find this to be particularly relevant for groundmotion verification and validation of simulationsbecause this seems to be a field where there isa significant lack of consensus on how to evalu-ate the accuracy of results. Therefore, establishingpost-processing standards on how to handle bothrecorded and simulated signals is an effort worthpursuing. This, in turn, will help constrain theuncertainties introduced when processing resultsand limit the analysis to the simulation parameters,models, and assumptions.6. ACKNOWLEDGEMENTSThis research was supported by the U.S. GeologicalSurvey (award G14AP00034), and the Southern Cali-fornia Earthquake Center (awards 14022, 14028, and14032). SCEC is funded by NSF Cooperative Agree-ment EAR-1033462 and USGS Cooperative AgreementG12AC20038. The SCEC contribution number for thispaper is 2095. The authors also thank the comments andsuggestions by the anonymous reviewers, which helpedimprove the final version of the paper.7. REFERENCESAnderson, J. G. (2004). “Quantitative measure of thegoodness-of-fit of synthetic seismograms.” Proc. 13thWorld Conf. Earthq. Eng., Int. Assoc. EarthquakeEng., Vancouver, British Columbia, Canada (August).Paper 243.Bielak, J., Graves, R. W., Olsen, K. B., Taborda, R.,Ramírez-Guzmán, L., Day, S. M., Ely, G. P., Roten,D., Jordan, T. H., Maechling, P. J., Urbanic, J., Cui,Y., and Juve, G. (2010). “The ShakeOut earthquakescenario: Verification of three simulation sets.” Geo-phys. J. Int., 180(1), 375–404.Graves, R. W. and Pitarka, A. (2010). “Broadbandground-motion simulation using a hybrid approach.”Bull. Seis. Soc. Am., 100(5A), 2095–2123.Khoshnevis, N. and Taborda, R. (2014a). “Sensitivityof ground motion simulation validation criteria to fil-tering.” Abstr. SSA Annu. Meet., Anchorage, Alaska,April 30 – May 2.Khoshnevis, N. and Taborda, R. (2014b). “Sensitivityof ground motion simulation validation to signal pro-cessing and GOF criteria.” Proc. SCEC Annu. Meet.,number EEII 055, Palm Springs, CA, September 6–10.Oppenheim, A. V., Schafer, R. W., and Buck, J. R.(1989). Discrete-Time Signal Processing. PrenticeHall, Englewood Cliffs, 2 edition.Taborda, R. and Bielak, J. (2013). “Ground-motion sim-ulation and validation of the 2008 Chino Hills, Cali-fornia, earthquake.” Bull. Seis. Soc. Am., 103(1), 131–156.8

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