Seismic Wave Propagation in a Very Soft Alluvial Valley: Sensitivity

to Ground-Motion Details and Soil Nonlinearity, and Generation

of a Parasitic Vertical Component

by F. Gelagoti, R. Kourkoulis, I. Anastasopoulos, T. Tazoh, and G. Gazetas

Abstract This paper explores the sensitivity of 2D wave effects to crucial problemparameters, such as the frequency content of the base motion, its details, and soilnonlinearity. A numerical study is conducted, utilizing a shallow soft valley as a testcase. It is shown that wave focusing effects near valley edges and surface wavesgenerated at valley corners are responsible for substantial aggravation (AG) of theseismic motion. With high-frequency seismic excitation, 1D soil amplification is pre-vailing at the central part of the valley, while 2D phenomena are localized near theedges. For low-frequency seismic excitation, wave focusing effects are overshadowedby laterally propagating surface waves, leading to a shift in the location of maximumAG toward the valley center. If the response is elastic, the details of the seismicexcitation do not seem to play any role on the focusing effects at valley edges, butmake a substantial difference at the valley center, where surface waves are dominant.The increase of damping mainly affects the propagation of surface waves, reducingAG at the valley center, but does not appear to have any appreciable effect at the valleyedges. Soil nonlinearity may modify the 2D valley response significantly. For ideal-ized single-pulse seismic excitations, AG at the valley center is reduced with increas-ing nonlinearity. Quite remarkably, for real multicycle seismic excitations AG at thevalley edges may increase with soil nonlinearity. In contrast to the vertical componentof an incident seismic motion, which is largely the result of P waves and is usually oftoo high frequency to pose a serious threat to structures, the valley-generated parasiticvertical component could be detrimental to structures: being a direct result of 2D wavereflections/refractions, it is well correlated and with essentially the same dominantperiods as the horizontal component.

Introduction

Although the surface response of 2D alluvial valley for-mations has been extensively investigated in the literature,research interest has mostly focused on valleys of idealizedgeometry (cosine-shaped, circular, elliptical, trapezoidal,etc.) subjected to idealized seismic motions (e.g., harmonicexcitation or simple wavelets), assuming elastic soilresponse. Such analyses have provided deep understandingof the complicated wave propagation phenomena. Amongvarious valuable insights and findings, it was concluded thatsurface waves generated at the valley boundaries (Lovewaves when the excitation is SH waves; Rayleigh wavesin case of SV and P waves) propagate back and forth alongthe valley surface resulting in significant amplifications (Tri-funac, 1971; Wong and Trifunac, 1974; Bard and Bouchon,1980; Harmsen and Harding, 1981; Othuki and Harumi,1983; Aki, 1988; Todorovska and Lee, 1991; Fishmanand Ahmad, 1995). Hereafter, the term aggravation (AG) will

be used to indicate the severity of amplification of the motionabove what the 1D theory would predict.

Although the research on the subject has been extendedto 3D valley response (Sánchez-Sesma et al., 1989; Sánchez-Sesma and Luzón, 1995; Bao et al., 1996; Bielak et al., 1999,2000), the effects of soil nonlinearity have received limitedattention. In a pioneering study, Zhang and Papageorgiou(1996) studied with the nonlinear response of the MarinaDistrict during the Loma Prieta earthquake and showed thatwave focusing effects and lateral interferences graduallydiminish with increasing soil nonlinearity.

Lately, the critical issue of capturing the real aggravationmechanisms and the necessity to confirm the theoreticalresults has led to the development of fully instrumented testsites, which serve as large-scale natural laboratories. TheEuroseistest in the Volvi basin in Greece (Chávez-Garcíaet al., 2000; Raptakis et al., 2000; Makra et al., 2001;

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Bulletin of the Seismological Society of America, Vol. 100, No. 6, pp. 3035–3054, December 2010, doi: 10.1785/0120100002

Pitilakis, 2004; Makra et al., 2005), the Japanese seismo-graph arrays in Ashighara Valley (Ohtsuki and Harumi,1983; Ohtsuki et al., 1984) and Ohba Valley (Tazoh et al.,1988; Gazetas et al., 1993), the alluvial Valley of Parkway inNew Zealand (Chávez-García et al., 1998), the CoachellaValley in California (Field, 1996), and the Valley of Nice inFrance (Sanchez-Sesma et al., 1988), are some of the bestknown test sites. Their merits include: (1) the high densityof the installed accelerograph arrays, (2) the detailed knowl-edge of subsoil geometry and soil mechanical properties,and (3) the accumulation of records. Site response analysisconfirmed the importance of 2D geometry effects, clearlysuggesting that 1D soil amplification phenomena may besignificantly contaminated (aggravated) by laterally propa-gating surface waves. Although such studies have offeredvaluable insights, in most cases only weak ground motionshave been recorded so far.

Despite the extensive bibliography on the subject, mostof the research conducted until now has focused on elasticsoil response and idealized input motions. The scope of thispaper is to gain further insight on the sensitivity of 2D waveeffects to crucial parameters, such as: (1) the frequency con-tent of the input motion, (2) the details of the input motion(duration, number of cycles, frequency content, etc.), and(3) soil nonlinearity. A numerical study is conducted, utiliz-ing the Ohba Valley (Japan) as an illustrative example. Inaddition, emphasis is given to the generation of parasiticvertical component, the effects of which may be detrimentalfor overlying structures, a phenomenon which has so farreceived scarce attention.

Problem Definition and Analysis Methodology

The Ohba Valley

Situated close to Fujisawa City in Japan, the Ohba Val-ley is an extremely soft alluvial basin. The valley is crossedby a 600 m long road bridge: Ohba Ohashi. The geometry ofthe valley and the soil profile are shown in Figure 1 (adaptedfrom Tazoh et al., 1984). The top layers (20 to 25 m) consistof extremely soft Holocene alluvium (organic layers ofhumus and clay). Despite the extensive soil improvementthat was conducted for the construction of the bridge, theNSPT values of the standard penetration test are very closeto zero, while the shear wave velocity, VS, measured throughdown-hole tests, ranges between 40 and 65 m=s.

The underlying substratum consists of Pleistocene dilu-vial deposits with NSPT values greater than 50 and VS around400 m=s. The groundwater table is almost at the ground sur-face, while the water content of the top layers by far exceeds100%. The latter are also characterized by large plasticityindex (in excess of 150); therefore, it is likely to exhibit elas-tic behavior even under strong seismic shaking (Vucetic andDobry, 1991). For more details see Tazoh et al. (1984).

Numerical Analysis Method

The problem is analyzed in the time domain employingthe finite element (FE) method, assuming plane-strainconditions. The idealized geometry of the valley and theassociated configuration of the FE model are depicted inFigure 2. The soil is modeled with quadrilateral continuumelements, with a very fine discretization to ensure realisticrepresentation of the propagating waves. The valley depositis assumed homogeneous with VS � 60 m=s, while the shearwave velocity of the substratum is significantly higher:VS � 400 m=s. With mass densities of 1.4 and 1:9 Mg=m3,respectively, the impedance contrast between soil and base,ρ2VS2=ρ1VS1 is about 10.

Reflections at the base of the formation are avoidedby utilizing absorbing boundaries. Free-field boundaries re-sponding as shear beams are placed at each lateral boundaryof the model, to simulate the motion produced by in-planevertically incident SV waves.

Three different types of analysis are conducted: (1) vis-coelastic analysis, utilizing the finite element code ABAQUS(2008); (2) equivalent-linear analysis, utilizing the codeQUAD4M (Hudson et al., 1994); and (3) nonlinear analysiswith ABAQUS, employing a kinematic hardening constitu-tive model. By comparing the results of viscoelastic withnonlinear (equivalent linear and fully nonlinear) analyses,the effects of soil nonlinearity can be quantified.

Soil Constitutive Modeling

For the nonlinear analyses, a nonlinear kinematic hard-ening constitutive model is employed. The evolution law ofthe model consists of two components: a nonlinear kinematichardening component, which describes the translation of theyield surface in the stress space (defined through the back-stress α, a parameter which describes the kinematic evolutionof the yield surface in the stress space), and an isotropichardening component, which describes the change of theequivalent stress controlling the size of the yield surfaceσo as a function of plastic deformation.

The model incorporates a Von Mises failure criterion,considered adequate to simulate the undrained response ofclayey materials, with an associative plastic flow rule(Anastasopoulos et al., 2010). The evolution of stresses is de-scribed by the relation

σ � σo � α: (1)

The evolution of the kinematic component of the yield stress isdescribed as follows

_α � C1

σ0

�σ � α�_�εpl � γα_�εpl; (2)

where C is the initial kinematic hardening modulus (C �σy=εy � E) and γ is a parameter that determines the rateof kinematic hardening decrease with increasing plasticdeformation.

3036 F. Gelagoti, R. Kourkoulis, I. Anastasopoulos, T. Tazoh, and G. Gazetas

Model parameters are calibrated against G-γ curvesof the literature, as described in Gerolymos et al. (2005).Figure 3 illustrates the results of one such calibration

(through finite element simulation of the simple sheartest) against the G-γ curves of Ishibashi and Zhang(1993).

(a)

(b)

Figure 1. (a) Cross-section of the Ohba Valley at the location of Ohba Ohashi (after Tazoh et al., 1984) (vertical scale exaggerated); and(b) soil profile along with NSPT and in-situ measured shear wave velocity VS.

Figure 2. (a) Idealized cross-section of the Ohba valley modeled in the paper; and (b) finite element discretization, along with zoomedview at the edge of the valley.

Seismic Wave Propagation in a Very Soft Alluvial Valley 3037

Validation against Recorded Response

The numerical analysis methodology employed hereinhas been extensively validated against recorded seismicresponse in Tazoh et al. (1988), Fan (1992), and Psarropou-los et al. (2007). Among a number of recorded seismicevents, two earthquakes were selected for analysis: (1) anearthquake ofMJMA � 6:0 at 81 km epicentral distance, withrecorded PGA � 0:03 g at the ground surface, referred to asearthquake A; and (2) the MJMA � 6:0 earthquake at 42 kmepicentral distance, with recorded PGA � 0:12 g at theground surface, referred to as earthquake B.

A comparison between the FE computed ground motionat the valley surface with the recorded is reproduced inFigure 4, in terms of elastic acceleration response spectra,SA. Given the relatively small acceleration amplitude of bothearthquakes (0.029 to 0.114 g), the shaking-induced shearstrains within the soil will not be large enough to generateany substantial soil nonlinearity. In fact, the alluvial layers ofthe valley are characterized by large plasticity indexes (inexcess of 100), and are thus expected to behave almost lin-early, even for larger imposed strains. Hence, the problemwas analyzed assuming elastic soil response with dampingratios of ξ � 1% and ξ � 3%, respectively. Evidently, forboth earthquakes the comparison is quite satisfactory, captur-ing most of the features of valley response. Note that thesefeatures could not possibly be captured through 1D soilresponse analysis (i.e., ignoring 2D wave effects).

The Effect of Frequency Content

To investigate the frequency-dependent scatteringphenomena, we first utilize Ricker wavelets as seismicexcitations (Ricker, 1960). The displacement time historyof these wavelets is given by

u�t� � �1 � 2b�t � to�2�e�b�t�to�2 ; (3)

where the parameter b is defined as b � πf2o, fo is thecharacteristic frequency of the pulse, and to is the timefor which u�t� is maximized. In the sequel, three character-

istic frequencies are used to illuminate the effects of fre-quency content on the dynamic response of the valley:

• a high-frequency Ricker 3 (with fo � 3 Hz),• a low-frequency Ricker 0.5 (with fo � 0:5 Hz), and• an intermediate Ricker 1 (with fo � 1 Hz).

Figure 5 depicts the acceleration time histories of thethree idealized wavelets (all scaled to 0.20 g), along withtheir corresponding response spectra.

The following sections go through the key findings ofthis analysis. Results are shown in terms of peak groundacceleration (PGA) and wave-field patterns.

Spatial Distribution of Peak GroundAcceleration (PGA)

Figure 6 depicts the spatial distribution of the aggrava-tion factor AG � A2D=A1D (defined as the ratio of peakground accelerations from the 2D and 1D analyses) alongthe valley surface for the three Ricker wavelets. All resultsrefer to elastic analysis with a damping ratio of ξ � 2%.

In the case of the high-frequency Ricker 3 wavelet(Fig. 6a), 1D soil amplification is clearly prevailing at thecentral part of the valley (AG≈ 1), while strongly 2D phe-nomena are localized near the edges. At those areas, trappingof obliquely incident body waves tends to amplify themotion experienced near the edges, resulting in appreciable

Figure 3. Nonlinear kinematic hardening constitutive soil mod-el calibrated against published G-γ curves from the literature.

(a)

(b)

Figure 4. Validation of numerical analysis method againstobservations: comparison of spectral accelerations response SAderived from the accelerograms recorded at the ground surface(point S) and the base stiff soil (point B) with those derived fromthe FE computation (assuming viscoelastic soil response). Two dif-ferent earthquake motions (as recorded at B) were used as excitation(Psarropoulos et al., 2007).

3038 F. Gelagoti, R. Kourkoulis, I. Anastasopoulos, T. Tazoh, and G. Gazetas

aggravations (AG≈ 1:3). Such focusing effects have beenaddressed, among others, by Sanchez-Sesma et al. (1988).This particular aggravation pattern is reminiscent of thedistribution of damage observed in several earthquakes: inCaracas, for example, the high concentration of damage inthe area of Palos Grandes during the 1967 earthquake wasattributed to the steep slope of the underlying bedrock atthe northern boundary of the 3 km wide sendimentary valley

(Papageorgiou and Kim, 1991), rather than simply to thelarge thickness of the soil deposit and the ensuing 1D waveamplification.

As shown in Figure 6b, the decrease of the dominantfrequency of the input seismic motion (Ricker 1: fo �1 Hz) leads to a different distribution of AG along the valleysurface, with the maximum AG (on the order of 1.7) beingobserved closer to the center of the valley. Observe also therapid fluctuations of AG from point to point along the sur-face. Evidently, 2D phenomena associated with multiplyreflected waveforms at the slope of the bedrock, which weredominant in the case of the high-frequency Ricker 3 seismicexcitation, are now absorbed: the length of the wave hasbecome too large to be affected by the topographic anomaly(i.e., the slope of the supporting bedrock). Hence, sucheffects are clearly overshadowed by the laterally propagatingsurface waves, leading to a shift of the location of the max-imum AG toward the center of the valley. Conversely, the AGfactor drops even below 1.0 close to the valley edges.

Figure 6c depicts the distribution of AG along the valleysurface for the low-frequency Ricker 0.5 wavelet. In this case,basin-induced waves strongly contaminate the 1D valleyresponse triggering a strongly 2D behavior along the wholevalley length. The maximum observed aggravation reaches1.4 at the center of the valley. Observe that the distributionof AG is quite similar to the previous case (Ricker 1), withthe main difference being the absence of the previouslydiscussed fluctuations: the increase of the wavelength hasapparently increased the distance between those anomaliesand in effect smoothened them significantly.

It is believed that themaximum aggravation in themiddleof the valley is the result of the constructive interferenceof Rayleigh waves, generated at the valley edges and propa-gating horizontally along the surface in opposite directions.Furthermore, the interference of the directly arrivingvertically propagating SV wave pulse with the horizontallypropagating Rayleigh waves is responsible for the observedpeak values at x � �130 m. The absence of conspicuousfocusing effects is hardly surprising, given the large wave-length of the incident SV waves (on the order of 60 �m=s�=0:75 �s�1� � 80 m) compared with the dimensions of thebedrock irregularity. The previous remarks will be further jus-tified in the sequel by means of seismogram synthetics.

It must be generally noted that the symmetrical shapeof the valley undoubtedly plays a significant role as thediffracted waves reach the middle of the valley in phase.Any potential asymmetry of the valley geometry may signif-icantly modify the aggravation pattern.

Wave-field Patterns: Seismogram Synthetics

To get a deeper insight into the aggravation generationmechanisms, a useful numerical diagnostic tool is the seis-mogram synthetics. Figures 7 and 8 depict the synthetics ofhorizontal and vertical acceleration, respectively, along thevalley surface.

Figure 5. The three Ricker wavelets used as seismic excitations:(a) the high-frequency Ricker, fo � 3 Hz; (b) the low-frequencyRicker, f0 � 0:5 Hz; (c) the intermediate Ricker, fo � 2 Hz; alongwith (d) their 5% damped acceleration elastic response spectra.

Figure 6. The effect of frequency content on the response alongthe ground surface: elastic analysis with soil hysteretic dampingξ � 2%. Distribution of the aggravation factor AG for: (a) thehigh-frequency Ricker 3 wavelet, (b) the intermediate Ricker 1,and (c) the low-frequency Ricker 0.5.

Seismic Wave Propagation in a Very Soft Alluvial Valley 3039

In the case of the high-frequency Ricker 3 seismicexcitation (Figs. 7a and 8a), one can clearly observe thegeneration of laterally induced Rayleigh waves that propa-gate toward the middle part of the valley with their amplitudegradually decreasing due to damping. Recall that in this casethe aggravation factor in the central part of the valley is in-deed equal to about 1.0 (no Rayleigh wave interference).

For the intermediate Ricker 1 seismic excitation, the re-sulting wave-field patterns are presented in Figures 7b and8b. All the different waveforms are clearly depicted: bodywaves (SV), refracted inclined waves (Ca), and two differentmodes of Rayleigh waves (R1 and R2, respectively). The firstmode (denoted as R1 in the figure) travels at 120 m=s and isbelieved to be the mode with the significant horizontal

Figure 7. The effect of frequency content on ground-motion synthetics. Elastic analysis with soil hysteretic damping ξ � 2%. Wavefields of horizontal acceleration at the ground surface of the valley for: (a) the high-frequency Ricker 3 wavelet, (b) the intermediate Ricker 1,and (c) the low-frequency Ricker 0.5.

3040 F. Gelagoti, R. Kourkoulis, I. Anastasopoulos, T. Tazoh, and G. Gazetas

component. The mode with the prevailing vertical behavior(clearly seen in the seismograph synthetic of the verticalmotion in Fig. 8b) propagates with significantly lowervelocity (65 m=s), an observation that agrees completelywith theoretical expectations. The refracted inclined wavespropagate along the horizontal x axis with an apparentpropagation velocity Ca defined as

Ca � VS= sinψ≈ 60=0:287≈ 209 m=s; (4)

where VS � 60 m=s and tanψ � 24=80 the slope inclina-tion. This theoretical value of velocity agrees fairly well withthe one graphically measured (≈200 m=s).

In case of low-frequency Ricker 0.5 (Figs. 7c and 8c),Rayleigh waves generated at both edges of the valley and pro-pagating toward its center are clearly illustrated (denoted asR1 in the figure). Their graphically measured velocity is equalto about 100 m=s, which is in good accordwith the theoretical

Figure 8. The effect of frequency content on ground-motion synthetics. Elastic analysis with soil hysteretic damping ξ � 2%. Wavefields of vertical acceleration at the ground surface of the valley for: (a) the high-frequency Ricker 3 wavelet, (b) the intermediate Ricker 1,and (c) the low-frequency Ricker 0.5.

Seismic Wave Propagation in a Very Soft Alluvial Valley 3041

value calculated based on the dispersion curve of Ohtsuki andHarumi (1983). Only one vibrating mode is stimulated. Notealso the collision of the opposite propagating Rayleigh wavesat the center of the valley. This fact confirms our former as-sumption that the high aggravations around the valley centerare attributed to Rayleigh wave constructive interference.

The Effect of the Details of the Seismic Excitation

In the previous section, some insights on the prevailingrole of frequency content to the resulting wave scatteringphenomena were investigated and discussed. The objectivehere is to examine whether, and to what extent, the largelyunpredictable details of the seismic excitation (duration,number of cycles, frequency content, etc.) influence the2D valley response.

To this end, the valley is subjected to real earthquakerecords. It is worth mentioning that in their majority the seis-mic motions used in the analysis have been recorded at thesurface of soil deposits, and therefore do not necessarily con-stitute realistic bedrock excitations. However, the scope ofthis analysis is not to predict the surface response of theexamined formation for a given seismic motion. Our inten-

tion is a systematic investigation of the problem, in order todelineate which input motion characteristics are responsiblefor the significant aggravation.

Three time histories will be analyzed (see the Data andResources section), each corresponding to a characteristicfrequency range: (1) the Kede record of the 1999 Ms 5.9Athens (Greece) earthquake (Papadopoulos et al., 2000;Gazetas et al., 2002); (2) the record of the 2003 Ms 6.4 Lef-kada (Greece) earthquake (Benetatos et al., 2005; Gazetaset al., 2005); and (3) the Yarimca record of the 1999 Mw 7.4Kocaeli (Turkey) earthquake (Elnashai, 2000). Because ofthe complexity of wave scattering phenomena with realrecords, seismogram synthetics are inadequate for the speci-fic analyses. Therefore, the results will be presented solely interms of peak ground accelerations.

High-Frequency Seismic Excitation: Kede(Greece) 1999

Figure 9a compares the time history of accelerationand the elastic response spectra of the Kede record withthe Ricker 3 wavelet (both high-frequency seismic excita-tions). Evidently, the comparison between the real record

(a)

(b)

Figure 9. The effect of the details of the seismic excitation. Elastic analysis with soil hysteretic damping ξ � 2%. (a) Acceleration timehistories and elastic response spectra of the Kede (Athens, 1999) record, compared with a fitted idealized Ricker wavelet (fo � 3 Hz);(b) comparison of the aggravation factor AG along the ground surface of the valley for the Kede (Athens, 1999) record and the idealizedRicker wavelet (Ricker 3).

3042 F. Gelagoti, R. Kourkoulis, I. Anastasopoulos, T. Tazoh, and G. Gazetas

Figure 10. The effect of the details of the seismic excitation. Elastic analysis with soil hysteretic damping ξ � 2%. (a) Acceleration timehistories and elastic response spectra of the Lefkada 2003 record, compared with a fitted idealized Ricker pulse (fo � 1:5 Hz); (b) compar-ison of the aggravation factor AG along the ground surface of the valley for the Lefkada 2003 record and the Ricker 1.5 pulse.

Figure 11. The effect of the details of the seismic excitation. Elastic analysis with soil hysteretic damping ξ � 2%. (a) Acceleration timehistories andelastic response spectra of theYarimca1999 record, comparedwith a fitted idealizedRickerwavelet (fo � 0:5 Hz); (b) comparisonof the aggravation factor AG along the ground surface of the valley for the Yarimca 1999 record and the idealized Ricker wavelet (Ricker 0.5).

Seismic Wave Propagation in a Very Soft Alluvial Valley 3043

and the idealized pulse is quite favorable. Apart from certainirregularities observed in the record, the time history and thefrequency content (see SA) of the two motions are rathersimilar.

The distribution of the aggravation factor AG alongthe ground surface for the two motions are compared inFigure 9b. At the valley edges, the agreement between realrecord and Ricker pulse is quite remarkable. Not only thedistribution pattern, but also the peak values of AG arepractically the same. However, moving toward the centerof the valley, the two distributions start exhibiting significantdiscrepancies: two more peaks of AG (at x≈�200 m and�150 m) appear with the Kede record; these are not ob-served with the Ricker 3 excitation. Still though, the behaviorin the central part of the valley is in both cases practically 1D,with the maximum AG being about 1.0.

Intermediate Seismic Excitation: Lefkada(Greece) 2003

With a rather large number of strong motion cycles (onthe order of 8), it could be argued that the record of theLefkada 2003 earthquake is one of the worst seismic motionsever recorded in Greece. The acceleration time history of therecord (characterized as an intermediate seismic excitation)is compared in Figure 10a with the idealized fitted Ricker 1.5pulse, which exhibits practically the same frequency content.

The results of the numerical analysis (always in termsof distribution of AG along the valley surface) are summar-ized in Figure 10b. Observe that the AG for the Lefkada2003 record is significantly higher compared with Rick-er 1.5, despite the similarity in frequency content. Noticealso that while the Ricker 1.5 generates a single, ratherdistinct peak of AG (at x≈�130 m), the real record ischaracterized by a large number of fluctuations: peaksatx≈�75 m and �20 m.

Interestingly, the response of the real record agrees fairlywell with that of the idealized pulse close to the valley edges.The irregularities of the record do not affect the response atthe valley edges, where focusing effects are dominant, butmake an important difference toward the center of the valley,where horizontally propagating Rayleigh waves seem to bein control. While the single pulse of the Ricker waveletcreates a single Rayleigh wave, the multiple strong motionpulses of the record are responsible for the development ofa multitude of surface waves. Obviously, the increase of thenumber of such waves increases the probability of construc-tive interference at different locations as they travel towardthe center of the valley.

Low-Frequency Seismic Excitation: Yarimca(Kocaeli) 1999

At a distance of only 3 km from the North Anatolianfault (responsible for the Kocaeli 1999 earthquake), theYarimca record is characterized by both forward-rupturedirectivity and fling-step effects (Garini et al., 2010). Once

the directivity and fling pulses are unveiled, the record(Fig. 11a) appears to be comparable with the Ricker 0.5wavelet in terms of frequency content. The comparison iscertainly not perfect in terms of SA, but Ricker 0.5 can beseen to reasonably fit the (first at least) hidden low-frequencyacceleration pulse of the record (see acceleration time his-tories). Naturally, the record is in addition characterizedby subsequent low-frequency pulses and multiple higher-frequency perturbations, which are also evident in the elasticresponse spectra, (observe the higher-frequency peaks).

As shown in Figure 11b, despite the substantial differ-ences between the record and Ricker 0.5, the agreementamong the distributions of AG is quite remarkable. Some dis-crepancies between the record and the idealized pulse doexist, but the general trend is quite similar. The record yieldsslightly higher maximum AG and is characterized by a moreirregular distribution. It could be claimed that the aforemen-tioned hidden low-frequency acceleration pulse of the recordyields a distribution of AG almost identical to that of theRicker 0.5, while the higher-frequency irregularities areresponsible for the observed fluctuations:

(a)

(b)

(c)

Figure 12. The effect of damping ratio–elastic analysis. Distri-bution of the aggravation factor AG along the valley surface forξ � 2%, 5%, and 10% for: (a) the high-frequency Ricker 3 wavelet,(b) the intermediate Ricker 1, and (c) the low-frequency Ricker 0.5.

3044 F. Gelagoti, R. Kourkoulis, I. Anastasopoulos, T. Tazoh, and G. Gazetas

• constructive interference of Rayleigh surface waves at theareas of AG local peaks (x≈�200 m and �100 m);

• destructive interference at the areas of AG local troughs(x≈�250 m and �160 m).

At this point, it should be noted that the comparisonbetween Ricker pulses (i.e., narrow band seismic motions)and real records (i.e., broadband seismic motions) wouldnot necessarily be equally acceptable if the soil was nothomogeneous.

The Effect of Soil Nonlinearity

The role of nonlinear soil response is investigated inthree different ways: (1) with viscoelastic analyses, in whicha small degree of soil nonlinearity is partially accounted forthrough increased damping ξ; (2) with equivalent-linearanalysis (in which a moderate degree of soil nonlinearityis taken into account through an iterative procedure accord-ing to which the soil stiffness G and the damping ratio of ξare made consistent with the shear strain level); and (3) withfully nonlinear analysis, in which strongly nonlinear soil

response is taken into account with the aforementionedkinematic hardening constitutive model.

The Influence of Damping Ratio

All of the results shown until now referred to elasticanalysis with ξ � 2%, an assumption that can be considered(a)

(b)

(c)

Figure 13. The effect of soil nonlinearity: comparison ofequivalent linear with nonlinear analysis using a kinematic harden-ing constitutive model. Distributions of the aggravation factor AGalong the valley surface for: (a) the high-frequency Ricker 3 wave-let, (b) the intermediate Ricker 1, and (c) the low-frequencyRicker 0.5 (PGA � 0:2 g for all input motions).

Figure 14. Comparison of equivalent linear with fully non-linear analysis: distributions of horizontal peak ground accelerationfor the Ricker 1 wavelet with PGA � 0:2 g.

(a)

(b)

(c)

Figure 15. The effect of soil nonlinearity: comparison of vis-coelastic with nonlinear (with the kinematic hardening constitutivemodel) analyses using real records as seismic excitation. Distribu-tions of the aggravation factor AG along the valley surface for:(a) the Kede, Athens 1999 record; (b) the Lefkada 2003 record;and (c) the Yarimca, Kocaeli 1999 record.

Seismic Wave Propagation in a Very Soft Alluvial Valley 3045

valid for (very) small magnitude seismic excitation and/orvery stiff soil. At such low shear strain amplitudes, the secantshear modulus G is very close to the initial (elastic) shearmodulus Gmax. However, with stronger seismic motions, thesoil will behave nonlinearly: G will decrease with increas-ing amplitude of shear strain, and the damping ratio willincrease. The scope of this section is to reveal whether andto what extent material damping influences the dynamic

response of the valley. For this purpose, the analyses arerepeated, with parametrically varying ξ between 2% and10%. To keep comparisons simple, results are discussedfor the three idealized Ricker pulses only.

Figure 12 summarizes the results in terms of AG distribu-tion along the valley surface. A general conclusion is that theincrease of the damping ratio ξ mainly influences surfacewave propagation. Observe that the local peaks toward the

(a) (b)

(c) (d)

(d)

(e)

Figure 16. Generation of parasitic vertical component–elastic analysis (ξ � 2%) using the horizontal component of the Kede, Athens,1999 recorded as (a), (b) the horizontal and vertical acceleration time histories at the points A and B of the valley surface; (c) the distributionof the ratio of vertical to horizontal acceleration component along the valley surface; and (d) the sole seismic excitation.

3046 F. Gelagoti, R. Kourkoulis, I. Anastasopoulos, T. Tazoh, and G. Gazetas

center of the valley, which are related to Rayleigh waveinterferences, decrease substantially with the increase of ξ.In contrast, the increase of ξ does not appear to have any effectonAGat thevalley edges.Hence, for the high-frequencyRick-er 3 (Fig. 12a), the increase of ξ does not appear to have anyeffect on the distribution of AG. Recall that in this case theaggravation is purely related to focusing effects, which arethe result of multiple wave reflections at the ground surfaceand the sloping bedrock. This mechanism, the direct result ofgeometry, is naturally not affected by the damping ratio. Onthe other hand, the aggravation due to surface waves requiresthat these waves, generated at the edges, propagate and reachthe center of the valley. Hence, since the increase of ξ tends tosubstantially dampen their propagation, the related aggrava-tion unavoidably decays as well.

This phenomenon becomes more evident in the case ofthe intermediate Ricker 1 wavelet (Fig. 12b), in which caseAG at x � 0 m (which is clearly related to constructive inter-ference of surface waves) reduces from 1.65 for ξ � 2% to

roughly 1.0 for ξ � 10%. Observe that the geometry-relatedAG at the valley edges is again insensitive to increasing ξ.The conclusions are qualitatively similar for the low-frequency Ricker 0.5 (Fig. 12c). Analyses with real records,not shown here for the shake of brevity, lead practically to thesame conclusions.

Equivalent Linear versus Fully Nonlinear Analysis

In this section, the results of equivalent linear analysis(using the numerical code QUAD4M) are compared withthose of a fully nonlinear analysis employing a kinematichardening constitutive model (see the detailed descriptionpreviously stated). In the first case, the analysis is practicallyelastic, but soil nonlinearity is taken into account through aniterative procedure according to which the soil stiffness Gand the damping ratio ξ are made consistent with the shearstrain level. In the latter case, nonlinear soil response is mod-eled with an increased degree of realism. For the equivalentlinear analysis, theG-γ curves of Ishibashi and Zhang (1993)

(a)

(b)

(c)

Figure 17. Vertical acceleration time histories and elastic response spectra at the valley surface, (a) at point A, and (b) at point B,compared with (c) vertical acceleration time history of the Kede, Athens 1999 record and corresponding elastic response spectra. Inthe SA diagrams, the respective horizontal acceleration time histories are also plotted (dashed lines) to allow for direct comparison.

Seismic Wave Propagation in a Very Soft Alluvial Valley 3047

for PI � 50 have been utilized. For the nonlinear analysis,the same curves are employed for calibration of constitutivemodel parameters (see Fig. 3). As in the previous section,results are shown for the three Ricker wavelets, all scaledat PGA � 0:2 g.

The comparison is summarized in terms of distributionof AG along the valley surface in Figure 13. Although thegeneral trends can be claimed to be comparable, the genericconclusion is that the two methods may yield differentresults.

In the case of the high-frequency Ricker 3 (Fig. 13a),although the maximum AG (≈1:4) predicted by the twomethods is quite similar, their distributions have noticeablediscrepancies. It is interesting to notice the shift in the loca-tion of the maximum AG: from x≈�250 m for the equiva-lent linear analysis (denoted with the gray line) tox≈�220 m for the nonlinear analysis. Going back to the

elastic analysis (see Fig. 12a), it becomes clear that boththe location and the amplitude of maximum AG producedby the equivalent linear analysis is almost the same as thatof the elastic analysis. This is attributable to the high fre-quency of the seismic excitation, due to which the developedshear strain is not enough to mobilize a large degree ofnonlinearity.

Conversely, in the cases of both the intermediate Rick-er 1 (Fig. 13b) and the low-frequency Ricker 0.5 (Fig. 13c),the induced nonlinearity practically eliminates the 2D aggra-vation phenomena previously attributed to Rayleigh waves.This trend is captured by both the equivalent linear and thefully nonlinear model, with the former predicting quite high-er values of AG.

Figure 14 compares the distribution of peak horizontalaccelerations along the valley surface (for the case of Ricker 1excitation) computed by means of equivalent linear and fully

Figure 18. Generation of parasitic vertical component–elastic analysis (ξ � 2%) using the horizontal component of the Lefkada 2003recorded as (a), (b) the horizontal and vertical acceleration time histories at the ground surface at point A; (c) the distribution of the ratio ofvertical to horizontal acceleration component along the valley surface; and (d) the sole seismic excitation.

3048 F. Gelagoti, R. Kourkoulis, I. Anastasopoulos, T. Tazoh, and G. Gazetas

nonlinear analysis, in order to demonstrate the very goodagreement of the two methods at the central part of the valley,where the response is dominated by 1D soil amplification.Any differences are localized at valley edges, where 2D wavescattering phenomena determine the response. Results aresimilar for all three Ricker wavelets.

Fully Nonlinear versus Elastic Analysis

Having investigated the differences between equivalentlinear and nonlinear analysis, the latter is employed in thissection to investigate the role of soil nonlinearity for realseismic excitations.

The comparison of elastic (ξ � 2%) with nonlinear ana-lysis is shown in Figure 15 in terms of distribution of AGalong the valley surface. Quite interestingly, and contraryto the common expectation, it appears that soil nonlinearitydoes not always cause AG to reduce. In fact, for the high-frequency Kede (Athens 1999) seismic excitation (Fig. 15a),AG at the valley edges increases with soil nonlinearity (fromroughly 1.25 to 1.6). At the valley center, there is practicallyno difference.

The same observation is valid for the intermediateLefkada 2003 (Fig. 15b) and the low-frequency Yarimca(Fig. 15c): Aggravation at the valley edges increases whensoil nonlinearity is modeled. To explain this observation, thefollowing hypothesis is made: soil plastification near the soil-rock interface, leads to the formation of a very soft plastifiedzone. In the case of the single-pulse Ricker wavelets, soilplastification acted as a damping mechanism, leading to

reduction of AG. But in the case of real seismic excitations,which contain a large number of strong motion cycles, thepicture is altered: the zone of plastification is generatedby the first arriving waves (due to the initial strong motioncycles), and then acts as a trap for forthcoming (due to thesubsequent strong motion cycles) inciting waves. The latterare trapped in a narrow band between the plastic zone and thesurface, and are thus generating larger AG.

If the previously stated hypothesis holds true, then thisphenomenon should become more evident with the increaseof strong motion cycles. Indeed, the difference betweenelastic and nonlinear analysis (always referring to valleyedges) is larger for the Lefkada 2003 and the Yarimca seis-mic excitations that contain several strong motion cycles:AG≈ 1:65 for the nonlinear analysis compared to roughly1 (no amplification) for elastic analysis.

In the case of the intermediate frequency and multicycleLefkada 2003 seismic excitation, the fluctuations of AGtoward the valley center practically disappear with soilinelasticity. The nonlinearity increases the effective damping,which is of hysteretic nature in this case, reducing the aggra-vation related to laterally propagating surface waves (simi-larly to the previously discussed observations referring tothe increase of the damping ratio).

Generation of Parasitic Vertical Component

In the previous sections, the aggravation due to 2D val-ley effects has been investigated, focusing on the prevailinghorizontal component of the seismic motion. However, due

(a)

(b)

Figure 19. (a) Vertical acceleration time history and elastic response spectra at the valley surface (point A), compared with (b) recordedvertical acceleration time history of the Lefkada, 2003 record and the corresponding elastic response spectra.

Seismic Wave Propagation in a Very Soft Alluvial Valley 3049

to the geometry of the bedrock slope, a purely horizontalseismic motion will unavoidably generate a parasitic verticalcomponent. A first attempt to address such phenomena ispresented in the sequel, focusing on real records.

Figure 16 depicts the results for the high-frequencyKede seismic excitation. The analysis is conducted subject-ing the valley to the horizontal component of the recordonly (bottom). As revealed by the distribution of the ratio(maxAv=maxAh) of the valley-generated parasitic verticalcomponent Av to the horizontal component Ah (middle ofthe figure), a significant parasitic vertical component isdeveloped that may even exceed Ah close to the valley edges.Moreover, because the parasitic valley-generated Av is theresult of geometry, it is totally correlated with Ah (seeproduced surface acceleration time histories in Fig. 16a).

Figure 17 compares the natural recorded vertical com-ponent time history and spectrum with the parasitically gen-

erated ones at the valley surface. Observe that the verticalcomponent of the Kede record (Fig. 17c) is of higher fre-quency compared with the horizontal one. As with most realrecords, such high-frequency vertical components may notreally have any substantial effect on the performance ofstructures, even if completely correlated with the horizontalmotion (e.g., Fardis et al., 2003). On the other hand, the fre-quency content of the parasitic vertical component (Fig. 17a,b) is practically the same as that of the horizontal component,while its amplitude is dramatically higher than that of thenatural component. Hence, in stark contrast to the naturalvertical component, which is the result of P waves, thevalley-generated parasitic vertical component can be detri-mental for overlying structures.

For the intermediate Lefkada 2003 (Figs. 18, 19) and thelow-frequency Yarimca (Figs. 20, 21) seismic excitations,the results are not as intense (the ratio does not exceed

(a) (b)

(c)

(d)

Figure 20. Generation of parasitic vertical component–elastic analysis (ξ � 2%) using the horizontal component of the Yarimca,Kocaeli, 1999 recorded as (a), (b) the horizontal and vertical acceleration time histories at the ground surface at point A; (c) the distributionof the ratio of vertical to horizontal acceleration component along the valley surface; and (d) the sole seismic excitation.

3050 F. Gelagoti, R. Kourkoulis, I. Anastasopoulos, T. Tazoh, and G. Gazetas

0.65), but the key conclusion remains. Being mainly theresult of geometry (or focusing) effects, the parasitic verticalcomponent almost disappears at the center of the valley.

Figure 22 investigates the effect of nonlinearity on thegenerated parasitic vertical component. For the high-frequency Kede seismic excitation (Fig. 22a), the Av=Ah

ratio remains unaffected by the induced nonlinearity. Forthe intermediate Lefkada 2003 case (Fig. 22b) and the low-frequency Yarimca (Fig. 22c), the soil nonlinearity, while notaltering the general trend, modifies the Av=Ah ratio, espe-cially in the valley edges. The ratio appears even higherin the nonlinear case in these regions. This increase is ratherthe result of decreased Ah than increased Av.

Conclusions

A numerical study has been conducted, utilizing a shal-low soft valley as a test case, to gain insights on the sensi-tivity of 2D valley response on parameters such as thefrequency content of the input motion, its details, and soilnonlinearity. The numerical methodology employed hereinhas been validated against recorded seismic response. Thefollowing conclusions have emerged:

1. The dynamic response of the valley was shown to bestrongly 2D, and cannot possibly be captured through1D soil response analysis.

2. Wave focusing at the valley edges and surface wavesoriginating at the corners of the valley are responsiblefor substantial aggravation of the seismic motion.

3. In the case of high-frequency seismic excitation, 1D soilamplification is prevailing at the central part of thevalley (AG≈ 1), while strongly 2D phenomena arerestricted at the corners, where trapping of obliquelyincident body waves amplifies the motion, resultingin aggravation of (AG≈ 1:3).

4. For low-frequency seismic excitations, the wavelengthbecomes too large to be affected by the topographicanomaly (i.e., the slope of the supporting bedrock),and focusing effects are overshadowed by the horizon-tally propagating surface waves, leading to a shift of thelocation of the maximum AG toward the center of thevalley.

5. For elastic response, the details of the seismic excitationdo make a difference in the development of surfacewaves, responsible for the aggravation at the valley cen-ter while they do not affect to the same extent focusingeffects at valley edges (which are geometry related). Theincrease of the number of strong motion cycles increasesthe probability of constructive interference of surfacewaves traveling toward the center of the valley, thus in-creasing the resulting AG.

6. The increase of damping ξ mainly influences surfacewave propagation, reducing AG toward the center ofthe valley. Yet it does not appear to have any effecton AG at valley edges.

7. Soil nonlinearity may modify the 2D valley response to asubstantial extent. The equivalent linear method cancapture parts of the problem, but will not yield the sameresults as a fully nonlinear analysis.

Figure 21. (a) Vertical acceleration time history and elastic response spectra at the valley surface (point A), compared to (b) verticalacceleration time history of the Lefkada, 2003 record and the corresponding elastic response spectra.

Seismic Wave Propagation in a Very Soft Alluvial Valley 3051

8. For idealized single-pulse (Ricker) seismic excitations,soil nonlinearity in general reduces AG, mainly at thecenter of the valley (where the role of surface waves isdominant). At the valley edges, where the response iscontrolled by the geometry, the differences are not aspronounced.

9. The details of real seismic excitations complicate thingsfurther, and quite remarkably lead to an increase in AG at

the valley edges as soil nonlinearity increases. Soilplastification near the soil-rock interface at valley edges,leads to development of a very soft plastified zone: thisis generated by the first arriving waves, which act as atrap for incident waves, which are captured between theplastic zone and the surface, thus generating larger AG.

10. The 2D geometry of the valley (excited by exclusively-horizontal waves) generates a parasitic vertical

(a)

(b)

(c)

Figure 22. The effect of soil nonlinearity on the parasitic vertical component of motion. Comparison of viscoelastic with nonlinearanalysis using only horizontal components of real records as seismic excitation. Distributions of the ratio of vertical to horizontal accelerationalong the valley surface for: (a) the Kede, Athens 1999 record; (b) the Lefkada 2003 record; and (c) the Yarimca, Turkey, 1999, record.

3052 F. Gelagoti, R. Kourkoulis, I. Anastasopoulos, T. Tazoh, and G. Gazetas

component. Compared to the natural vertical componentof an earthquake, which is the result of P waves and isusually of very high-frequency content to pose a seriousthreat to structures, this valley-generated parasitic verti-cal component can be detrimental for overlying struc-tures: being a direct result of geometry, it is fullycorrelated and of practically the same dominant periodas the horizontal component.

Data and Resources

Seismograms used in this study were partially collectedby the online earthquake database cosmos vdc available atwww.cosmos‑eq.org/scripts/earthquakes.plx (last accessed15November 2009), and the Hellenic Accelerogram databasev 1.0 available at www.gein.noa.gr/HEAD (last accessed 15November 2009). Data can be ordered through the respectivewebpages. The valley soil profile data and geometry havebeen obtained from published sources listed in the references.

Acknowledgments

This work was partially supported by the EU 7th Framework researchproject funded through the European Research Council’s (ERC) IdeasProgramme, in Support of Frontier Research-Advanced Grant. Contractnumber ERC-2008-AdG 228254-DARE.

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National Technical University of Athens, Greece(F.G., R.K., I.A.)

Shimizu Corporation, Japan(T.T.)

National Technical University of Athens,Athens, Greece

(G.G.)

Manuscript received 4 January 2010

3054 F. Gelagoti, R. Kourkoulis, I. Anastasopoulos, T. Tazoh, and G. Gazetas