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

A fully parametric non-stationary spectral-based stochastic ground motion model Vlachos, Christos; Deodatis, George; Papakonstantinou, Konstantinos G. Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015A Fully Parametric Non-Stationary Spectral-Based StochasticGround Motion ModelChristos VlachosDoctoral Candidate, Dept. of Civil Engineering and Engineering Mechanics, ColumbiaUniversity, New York, NY, USAGeorge DeodatisProfessor, Dept. of Civil Engineering and Engineering Mechanics, Columbia University,New York, NY, USAKonstantinos G. PapakonstantinouResearch Scientist, Dept. of Civil Engineering and Engineering Mechanics, ColumbiaUniversity, New York, NY, USAABSTRACT: A novel strong ground motion stochastic model is formulated in association with physicallyinterpretable parameters that are capable of efficiently characterizing the complex evolutionary nature ofthe phenomenon. A multi-modal, analytical, fully non-stationary spectral version of the Kanai-Tajimimodel is introduced achieving a realistic description of the evolutionary spectral energy distribution of theseismic ground motions. The functional forms describing the temporal evolution of the model parameterscan effectively model complex highly non-stationary power spectral characteristics. The analysis space,where the analytical forms describing the evolution of the model parameters are established, is the energydomain instead of the typical use of the time domain. The Spectral Representation Method facilitates thesimulation of sample model realizations.1. INTRODUCTIONThe increased and constantly rising interest inperformance-based earthquake engineering, in par-allel to modern code requirements, has enhancedthe need for reliable, diverse and realistic groundmotion time-histories. Naturally, the availability ofactual seismic records pertaining to certain earth-quake scenarios can be proven of quite limited size.Engineers are thus frequently forced to scale and/ormodify the spectral content of actual records. Itis obvious that this procedure, mainly motivatedby necessity, is fraught with specific concerns re-garding the resulting representation of the groundmotions. Structural response is sensitive to inputloading characteristics and extreme care should begiven to an astute input modeling. Apart from over-coming the record scaling and/or spectral matchingconcerns, simulated earthquake waveforms basedon stochastic representations have the extra advan-tage that can be directly used for stochastic dy-namic analyses. In parallel to that, a descriptionof the seismic hazard in the form of simulatedwaveforms provides a meticulous characterizationof the seismic risk. Various stochastic ground mo-tion models have been formulated in the past, rang-ing from time-domain approaches, e.g. Rezaeianand Der Kiureghian (2008), to spectral-based, e.g.Deodatis and Shinozuka (1988); Conte and Peng(1997), and wavelet-based, e.g. Spanos and Failla(2004); Yamamoto and Baker (2013).In this paper, a new analytical non-stationaryspectral stochastic ground motion model is pre-sented in association with physically interpretableparameters. The model formulation is based ona novel multi-modal, fully non-stationary spectralversion of the Kanai-Tajimi model (Kanai (1957);Tajimi (1960)). In relation to all previous worksin the literature, the presented model is the first112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015fully analytical model to date that is capable of di-rectly and efficiently describing multi-modal evo-lutionary power spectral densities, allowing for arealistic description of the spectral energy distribu-tion over time. The introduced model and the de-veloped functional forms describing the temporalevolution of its parameters are capable of efficientlyrepresenting complex non-stationarities. The anal-ysis space, where the model’s analytical forms areestablished, is the energy domain, instead of thetypical use of the time domain. The model iscompletely defined in parametric form and sup-ports ground motion simulations through the Spec-tral Representation Method (Shinozuka and Deo-datis (1991); Deodatis (1996); Liang et al. (2007)).The parametric character of the model allows alsoits straightforward implementation in random vi-bration problems. By treating the model parame-ters as random variables and by examining a largeground motion database, the associated seismic riskcan be effectively quantified through identificationof the distribution patterns of the model parameters,in addition to estimating their correlation structure.In this way, the natural variability of the databaseground motions’ stochastic nature can be effec-tively captured. Such an analysis for a subset of theNGA West database can be found in detailed formin the under review paper by Vlachos et al. (2015).2. NON-STATIONARY SPECTRAL ESTI-MATION METHODThe fully non-stationary spectral estimation of theearthquake ground motion records is performed byemploying the Short-Time Multiple-Window esti-mation technique as formulated and presented ingreat detail by Conte and Peng (1997). Based on theassumption of local stationarity in the underlyingstochastic process, multiple orthogonal leakage-resistant moving time-windows are employed to ex-tract the spectral content of each signal segment.The resulting multiple local spectra are then av-eraged in a weighted sense to produce the non-stationary spectral estimate of the target process.The resulting non-stationary power spectrum is aconsistent estimator of the true spectrum, not ham-pered by the usual trade-off between bias and spec-tral leakage.3. EVOLUTIONARY MULTI-MODALSPECTRAL-BASED MODEL3.1. Evolutionary Bimodal Kanai-Tajimi ModelThe presented parametric, multi-modal, fully non-stationary model is based on the superpositionof classical unimodal Kanai-Tajimi expressions.Without any loss of generality, a bimodal version ispresented in this paper, which is deemed adequatefor the objectives of this work. The employmentof a high-pass filter is also necessary since it ap-propriately remedies the signal contamination dueto long-period noise in the simulated waveforms,e.g. Liao and Zerva (2006). The employed bimodalevolutionary model is thus expressed as:SXX ( f , t) =|HP( f )|2K=2∑k=1S(k)o (t)×1+(2ζ (k)g (t) f/ f (k)g (t))2(1−(f/ f (k)g (t))2)2 +(2ζ (k)g (t) f/ f (k)g (t))2(1)where SXX ( f , t) is the model evolutionary powerspectrum, HP( f ) is the deterministic Butterworthhigh-pass filter and{f (k)g (t) ,ζ (k)g (t) ,S(k)o (t)}are thetime-varying dominant modal frequency, modal ap-parent damping ratio and modal participation fac-tor with respect to the kth mode respectively. Themodal numbering is directly associated with the nu-merical value of the respective modal dominant fre-quencies, being ordered in an ascending way, i.e.f (1)g (t) is always the numerically smallest modalfrequency.3.2. High-Pass Butterworth FilterThe high-pass filter used in this study is a determin-istic high-pass (low-cut) Butterworth filter of 4th or-der. The energy content of the transfer function ofthe Butterworth filter is expressed as:|HP( f )|2 = ( f/ fc )2N1+( f/ fc )2N(2)where N = 4 indicates the order of the filter andfc its corner frequency. Following Liao and Zerva(2006), the corner frequency is selected in a purelynumerical way such that most of the low-frequencycomponents of the ground motion, correspondingto periods longer than the duration Td of the seismic212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015record, are practically eliminated by specifying alow amplitude energy threshold for the frequencyfd = 1/Td . The adopted energy threshold Hd is setto 10−3 uniquely defining the cut-off frequency as:fc =1Td(H2d/(1−H2d) )1/2N (3)where fc ∈ [0.10,0.25]Hz.3.3. Model Parameter IdentificationEmploying the Short-Time Multiple-Window non-stationary spectral estimation technique as outlinedin Section 2, in conjunction with the bimodal evo-lutionary Kanai-Tajimi model as in Eq. (1) and thedeterministic Butterworth filter as in Eqs. (2)-(3),allows for the estimation of the model parametersas shown below. The 6-parameter set x̂t defined as:x̂t ={̂f (1)g (t) , ̂ζ (1)g (t) , ̂S(1)o (t) , ̂f (2)g (t) , ̂ζ (2)g (t) , ̂S(2)o (t)}(4)is identified as the best estimator of the true param-eter time-varying set xt of the seismic record in thenon-linear least-squares sense. The minimizationis performed for every time instant over the entirefrequency domain and can be expressed as:x̂t = argminxt∫+∞−∞(S(s-t)XX ( f , t)−SXX ( f , t,xt))2d f(5)where S(s-t)XX ( f , t) is the evolutionary spectral esti-mate of the signal as obtained by the Short-TimeMultiple-Window technique and SXX ( f , t,xt) is themodel power spectrum as in Eqs. (1)-(3). It is im-portant to be noted here that the analytical formsdescribing the temporal evolution of the identifiedmodal characteristics are presented in the followingpages without any sort of hat accents.3.4. Transformation of Time-Domain to Energy-DomainOne of the innovative characteristics of the pre-sented study is that the analysis space, where theanalytical forms describing the temporal evolutionof the previously identified spectral characteristicsare established, as shown later, is the energy do-main instead of the typically employed time do-main. Based on this choice, the modeling of theevolving spectral content becomes more accurate inthe strong shaking part of the ground motion, wherethe greatest interest for engineering purposes lies.The adopted non-dimensional energy definition isanalogous to the Arias Intensity by Arias (1970)and is given by:ε (t) =∫ t0 x2 (τ)dτ∫ Td0 x2 (τ)dτ(6)where t ∈ [0,Td ] and ε (t) ∈ [0,1] is the non-dimensional cumulative energy contained in the[0,t] fraction of the seismic record x(t). In essence,Eq. (6) provides a non-linear mapping between thetime domain and the non-dimensional energy do-main, allowing higher resolution in the high ampli-tude part of the seismic ground motion.3.5. Analytical Forms Expressing Temporal Evo-lution of Model Parameters3.5.1. Dominant Modal FrequenciesThe evolutionary nature of the frequency contentof seismic ground motion records arises due to thenon-concurrent intermingling of the different typesof seismic waves at the site, as a direct result of theirdifferent propagating velocities. The body waves, Pand S waves, arrive faster compared to the surfacewaves, Rayleigh and/or Love waves, with the S-waves typically comprising the dominant portion ofthe strong motion part of the seismic records. Bodywaves are typically characterized by higher fre-quency content as compared to the surface waves,and are attenuated more as the source-to-site dis-tance increases. Furthermore, local site conditionscan significantly amplify the long period spectralenergy. As a result of this apparent complexity, themodal dominant frequencies cannot be accuratelycharacterized by only one specific temporal pattern,e.g. linear. Thus, the usually decaying nature of thedominant frequencies needs to be described by aversatile parametric expression being able of effec-tively describing different patterns of temporal vari-ation. The analytical form in this paper describingthe evolution of the two identified modal dominantfrequencies in the non-dimensional cumulative en-ergy domain is given as:f (k)g (ε) = Qk(12+ ε)αk(32− ε)βk(7)312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015for k = 1,2 and {Qk,αk,βk} being the parameters ofthe analytical form with respect to the kth mode,where αk,βk ∈ℜ and Qk > 0.In Fig. 1, the aforementioned fitted expres-sion is provided in colored thick line for bothmodes with respect to the Parkfield seismic record,along with the associated record identified values{̂f (1)g (ε) ,̂f (2)g (ε)}represented in thin black line.The used Parkfield seismic record is the compo-nent 320 of the Cholame-Shandon Array #8 record-ing (NGA Sequence No: 31) of the Parkfield 1966earthquake (Mw = 6.19). It is important to be notedhere that the entirety of the fitted functional formparameter values pertaining to the Parkfield recordcan be found in the Appendix.3.5.2. Modal Apparent Damping RatioConsidering the damping ratio as the signal band-width, there are observations according to whichthe signals exhibit a trend of having broader band-width at their beginning and end. On the otherhand, one may approach the apparent dampingas the equivalent viscous damping of a complexsystem experiencing material damping, which isknown to be amplitude dependent, therefore result-ing in higher apparent damping during the strongmotion part of the record. Furthermore, consideringthe high sensitivity of the damping ratio identifica-tion process and aiming to a sophisticated but alsoconcise model, the modal apparent damping ratiois considered a constant, however different for eachmode. It is eventually calculated as the averagedidentified damping ratio in the strong shaking partof the seismic record, with the latter being definedas the part of the ground motion between 5% and95% of the seismic record energy, given as:ζ (k)g (ε) = ζ (k)g ={̂ζ (k)g (ε)∣∣∣∣ε ∈ [0.05,0.95]}(8)for k = 1,2, where {} denotes the averaging opera-tor.3.5.3. Modal Participation FactorsThe identified modal participation factors{̂S(1)o (ε) ,̂S(2)o (ε)}are not directly associated0 0.2 0.4 0.6 0.8 101234567ε(t)f (Hz)Parkfield  f (1)g (ε) f (2)g (ε)Figure 1: Dominant modal frequencies fitting for theParkfield record.to the spectral energy carried by each mode andthus their numerical values mainly characterize theidentified linear combination of the modes, as inEq. (1). Consequently, their relative values are farmore important compared to their absolute ones,resulting in their normalization by the first modalparticipation factor. Accordingly, the logarithmicnormalized modal participation factors are definedas follows:S(k)o (ε) =̂S(k)o (ε)/̂S(1)o (ε)R̂(k) (ε) = log10[S(k)o (ε)] (9)for k = 1,2. It is apparent that the above mentioneddefinition leads to S(1)o (ε) = 1 and R̂(1) (ε) = 0 . Dueto the oscillatory character of the second logarith-mic normalized participation factor, its modelingneeds to be performed with a flexible parametric ex-pression. The chosen analytical form is a mixtureof two bell-shaped Gaussian functions, as follows:R(2) (ε) =F(I) exp−(ε−µ(I)σ (I))2+F(II) exp−(ε−µ(II)σ (II))2−2(10)where{F(I),F(II)}are the scaling factors,{µ(I),µ(II)}the peak locations and{σ (I),σ (II)}412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015parameters controlling the width of the ‘bells’.The scaling factors F and the width-controllingparameters σ are not allowed to take negativevalues. The labeling of this parametric set isdirectly associated with the numerical values of{µ(I),µ(II)}, assigning always the numericallysmaller of the two identified peak locations to µ(I)and the larger to µ(II). In Fig. 2, the fitted R(2) (ε)expression is provided for the Parkfield record,along with the associated record identified values,R̂(2) (ε), represented in thin black line.3.5.4. Energy Accumulation and Amplitude Mod-ulating FunctionThe use of the normalized form of the modal par-ticipation factors, as described in Eq. (9), results inthe non-uniform temporal modulation of the powerspectral intensity. In order to rectify the modulatedsignal intensity, the evolutionary power spectrumis converted to unit-variance at each time instant,through equating its integral over the entire fre-quency domain to unity, subsequently followed bythe introduction of an amplitude modulating func-tion z(t). The analytical form of the resulting fittedbimodal evolutionary power spectrum is then givenas:SXX ( f , t) = z2 (t)×|HP( f )|2 K=2∑k=1S(k)o (ε (t))1+(2ζ (k)g f/ f (k)g (ε (t)))2(1−(f/ f (k)g (ε (t)))2)2 +(2ζ (k)g f/ f (k)g (ε (t)))2︸ ︷︷ ︸unit-variance(11)In order to generally express the parameters ofEq. (11) in the time domain, a parametric formis required to describe the non-dimensional energyaccumulation over time, as defined in Eq. (6).The following non-dimensional analytical form isproven to effectively portray this measure, pertain-ing to the seismic records of the selected NGA Westdatabase:ε (t)=e−(t/Tdγ)−δe−(1γ)−δ (12)where {γ,δ} are positive scale and shape parame-ters respectively.It is also demonstrated that there is a remark-able connection between the amplitude modulat-0 0.2 0.4 0.6 0.8 1−2−1.5−1−0.500.511.5ε(t)Parkfield  R(2)(ε)Figure 2: Logarithmic normalized modal participationfactor R(2) (ε) fitting for the Parkfield record.ing function introduced in Eq. (11) and the en-ergy accumulation function described in Eq. (12).Consider the following slightly different definitionof the accumulated energy, adopting now a dimen-sional form, as follows:ε (t)Ix=e−(t/Tdγ)−δe−(1γ)−δ (13)where ε (t) ∈ [0, Ix] and Ix is the total energy contentof the seismic record x(t) defined as:Ix =∫ Td0x2 (τ)dτ (14)Equating the signal energy accumulation at any ar-bitrary time instant t ∈ [0,Td ] with the respective cu-mulative evolutionary spectral energy as follows:ε (t) =∫ t0∫ +∞−∞SXX ( f ,τ)d f dτ (15)and considering Eq. (11) results to:ε (t) =∫ t0z2 (τ)dτ (16)Finally, Eqs. (13) and (16) lead to the followingdefinition of the amplitude modulating function z(t)as:z2 (t)Ix= δ/γTd·e−(t/Tdγ)−δ(t/Tdγ)−1−δe−(1γ)−δ (17)512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20150 0.2 0.4 0.6 0.8 100.511.522.5x 104t/Tdε (cm2 /sec3 )Parkfield  ε(t)Figure 3: Energy accumulation function ε (t) fitting forthe Parkfield record.Hence, the modeling of the energy accumula-tion function serves two distinct and very importantgoals; the first one being the mapping between thetime and energy domains and the second one beingthe simultaneous formation of the required ampli-tude modulating function z(t), responsible for thetime modulated signal intensity. In Fig. 3, the fit-ted dimensional version of the energy ε (t) expres-sion of Eq. (13) is provided with respect to theParkfield record, along with the associated recordcumulative energy represented in thin black line.In Figs. 4–5, the evolutionary spectral estimateS(s-t)XX ( f , t) of the Parkfield record is provided, alongwith the associated fitted model evolutionary spec-trum SXX ( f , t) of Eq. (11) respectively. The evolu-tionary power spectra in the two figures are plottedin a non-dimensional normalized form for visual-ization purposes. The absolute magnitudes of thespectra are directly associated with the energy ac-cumulation function ε (t) fit, provided in Fig. 34. SIMULATION OF THE DEVELOPEDEVOLUTIONARY BIMODAL KANAI-TAJIMI POWER SPECTRUMFollowing the complete mathematical descriptionof the model, the Spectral Representation Method,Shinozuka and Deodatis (1991); Deodatis (1996);Liang et al. (2007), can be engaged in a straightfor-ward manner for the stochastic ground motion sim-ulations. Consider a zero-mean, real-valued non-t (sec)f (Hz)S(s-t)XX (f, t)  0 5 10 15 20 2505101500.20.40.60.81Figure 4: Estimated power spectrum S(s-t)XX ( f , t) for theParkfield record.t (sec)f (Hz)SXX (f, t)  0 5 10 15 20 2505101500.20.40.60.81Figure 5: Model evolutionary power spectrumSXX ( f , t) for the Parkfield record.stationary stochastic process x0 (t) with two-sidedevolutionary power spectrum Sx0x0 ( f , t), as in Eq.(11). It has been shown, Liang et al. (2007), thatthe stochastic process x0 (t) can be represented bythe following series as N→ ∞:x(t) =2N−1∑n=0[2Sx0x0 ( fn, t)pi∆ f ]1/2 cos(2pi fnt +Φn)(18)612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20150 5 10 15 20 25−300−200−1000100200300  time (sec)Acceleration (cm/sec2 )ParkfieldrecordFigure 6: Parkfield 1966, Cholame - Shandon Array #8(NGA No: 31), component 320.where a suitably discretized version of the evolu-tionary power spectrum is needed and:∆ f = fu/N , fn = n∆ f , Sx0x0 ( f0 = 0, t) = 0 (19)In Eq. (19), fu represents an upper cut-off fre-quency beyond which the evolutionary power spec-trum Sx0x0 ( f , t) may be assumed to possess negli-gible amount of power for either mathematical orphysical reasons. The {Φ0,Φ1,Φ2, ...,ΦN−1} in Eq.(18) are N independent random phase angles dis-tributed uniformly over the interval [0,2pi]. Thesimulated stochastic process x(t) is asymptoticallyGaussian as N→∞ due to the central limit theorem.Following the simulation of seismic accelerationtime-histories, velocity and subsequently displace-ment time-histories can be obtained via numericalintegration. However, the simulated accelerationtime-histories can be numerically contaminated,containing some amounts of low-frequency wave-form components, resulting in unrealistic velocitiesand displacements, both exhibiting increasing driftsas time increases and overestimating the structuralresponse in the low-frequency (long-period) range.Therefore, a simple post-processing filtering tech-nique is employed by causally applying the samehigh-pass Butterworth filter, as described in Sec-tion 3.2. A more detailed discussion about the usedfiltering scheme can be found in the under reviewpaper by Vlachos et al. (2015).0 5 10 15 20 25−300−200−1000100200300time (sec)Acceleration (cm/sec2 )  simulationFigure 7: Sample acceleration of the Parkfield record.In Figs. 6–7, the Parkfield seismic record is pro-vided, along with one filtered sample accelerationtime-history, respectively. The pseudo-acceleration(5% damped) elastic response spectrum of the Park-field record is provided (in thick line) in Fig. 8,accompanied by the response spectra from 100 re-alizations of the associated stochastic ground mo-tion model. As seen in Fig. 8, the actual responsespectrum of the Parkfield record is very well con-tained within the range defined by the responsespectral ordinates of the associated sample realiza-tions, evincing the efficiency of the suggested evo-lutionary spectral modeling.5. CONCLUSIONSA new analytical non-stationary spectral-basedstochastic ground motion model is presented inassociation with physically interpretable parame-ters. The new model is based on a novel multi-modal, non-stationary spectral version of the wellknown Kanai-Tajimi model and is the first fully an-alytical model to date that is capable of directlyand efficiently describing multi-modal evolution-ary power spectral densities, allowing for a real-istic description of the spectral energy distributionover time. The model, along with the functionalforms describing its time-varying parameters, de-pict high versatility in portraying distinctive highlynon-stationary power spectral characteristics. Theenergy domain is preferred over the time domainas the analysis space. Among others, the para-712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 201510−1 100 10110−310−210−1100Tn (sec)PSA (g)Parkfield  record samplesFigure 8: Pseudo-acceleration (5% damped) elasticresponse spectrum for the Parkfield record togetherwith 100 response spectra model realizations.metric character of the model allows its straight-forward implementation in random vibration prob-lems. By treating the model parameters as randomvariables and by examining a large ground motiondatabase, the associated seismic risk can be effec-tively quantified, capturing the natural variabilityof the database seismic motions. Furthermore, thefully analytical nature of the model is fitting for fu-ture development of appropriate predictive modelsthat can eventually link the complete evolutionarypower spectra to very specific descriptions of ‘site-based’ earthquake scenarios. The clear theoreticalconnection of the power spectra to different seismo-logical scenarios encourages this direction furtherand future works by the authors will be also con-centrated on this effort that can offer new knowl-edge and resources in the broad context of earth-quake engineering and seismic hazard analysis.APPENDIXTable 1: Model parameter values of Parkfield record.γ δ Q1 α1 β1 Q2 α2 β20.14 2.77 2.43 2.00 2.00 5.73 2.38 2.82ζ (1)g ζ (2)g F(I) µ(I) σ (I) F(II) µ(II) σ (II)0.16 0.21 1.11 0.06 0.08 2.74 0.39 0.59REFERENCESArias, A. (1970). “A measure of earthquake intensity.”Seismic Design for Nuclear Power Plants, The M.I.T.Press, 438–483.Conte, J. and Peng, B. (1997). “Fully nonstationary ana-lytical earthquake ground-motion model.” Journal ofEngineering Mechanics, 123(1), 15–24.Deodatis, G. (1996). “Non-stationary stochastic vec-tor processes: seismic ground motion applications.”Probabilistic Engineering Mechanics, 11(3), 149–167.Deodatis, G. and Shinozuka, M. (1988). “Auto-regressive model for nonstationary stochastic pro-cesses.” Journal of Engineering Mechanics, 114(11),1995–2012.Kanai, K. (1957). “Semi-empirical formula for the seis-mic characteristics of the ground.” Bulletin of Earth-quake Research Institute, Tokyo University, 309–325.Liang, J., Chaudhuri, S. R., and Shinozuka, M. (2007).“Simulation of nonstationary stochastic processes byspectral representation.” Journal of Engineering Me-chanics, 133(6), 616–627.Liao, S. and Zerva, A. (2006). “Physically compli-ant, conditionally simulated spatially variable seis-mic ground motions for performance-based design.”Earthquake Engineering & Structural Dynamics,35(7), 891–919.Rezaeian, S. and Der Kiureghian, A. (2008). “A stochas-tic ground motion model with separable temporal andspectral nonstationarities.” Earthquake Engineering& Structural Dynamics, 37(13), 1565–1584.Shinozuka, M. and Deodatis, G. (1991). “Simulation ofstochastic processes by spectral representation.” Ap-plied Mechanics Reviews, 44(4), 191–204.Spanos, P. D. and Failla, G. (2004). “Evolutionary spec-tra estimation using wavelets.” Journal of Engineer-ing Mechanics, 130(8), 952–960.Tajimi, H. (1960). “A statistical method of determiningthe maximum response of a building structure duringan earthquake.” Proc. of the 2nd WCEE, 2, 781–798.Vlachos, C., Papakonstantinou, K. G., and Deodatis,G. (2015). “A multi-modal analytical non-stationaryspectral model for characterization and stochasticsimulation of earthquake ground motions.” Soil Dy-namics and Earthquake Engineering, Submitted.Yamamoto, Y. and Baker, J. W. (2013). “Stochasticmodel for earthquake ground motion using waveletpackets.” Bulletin of the Seismological Society ofAmerica, 103(6), 3044–3056.8

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