Abstract We study the geometrical and material conditions which lead to focusing of seismicwaves traveling across a concave velocity interface representing the boundary of a sedimen-tary basin within a denser rock. We approximate, using geometrical analysis for plane-waves,the combination of interface eccentricities and velocity ratios for which the seismic rays con-verge to a near surface region of the basin. 2-D finite difference modeling is used to computePeak Ground Velocity (PGV) and spectral amplification across the basin. We show that effec-tive geometrical focusing occurs for a narrow set of eccentricities and velocity ratios, whereseismic energy is converged to a region of ±0.5 km from surface. This mechanism leads tosignificant amplification of PGV at the center of the basin, up to a factor of 3; frequenciesof the modeled spectrum are amplified up to the corner frequency of the source. Finally, wesuggest a practical method for evaluating the potential for effective geometrical focusing insedimentary basins.
Electronic supplementary material The online version of this article (doi:10.1007/s10518-013-9526-4)contains supplementary material, which is available to authorized users.
S. Shani-Kadmiel (B)Department of Geological and Environmental Sciences, Ben Gurion University of the Negev,Beersheba, Israele-mail: [email protected]
M. TsesarskyDepartment of Structural Engineering, Ben Gurion University of the Negev, Beersheba, Israel
J. N. LouieNevada Seismological Laboratory, University of Nevada, Reno, USA
Z. GvirtzmanGeological Survey of Israel, Jerusalem, Israel
Seismic ground motions may be adversely amplified atop sedimentary basins. In MexicoCity, Los Angeles, Kobe and other large urban centers located over large sedimentary basins,violent ground motions and prolonged shaking caused considerable structural damage andloss of life. Ground motion amplitude is known to be affected by the near surface materialproperties but in a number of cases, localized damage has been attributed to the internal,deeper structure of the basin. Evidence from ground shaking and site response in the LosAngeles basin, following the 1994 Northridge earthquake (Gao et al. 1996; Hartzell et al.1997; Graves et al. 1998) and from Kobe, following the 1995 Great Hanshin earthquake(Hatayama et al. 1995; Kawase 1996) emphasized the importance of the deep basin structure.Specifically, the focusing of seismic energy by geological structures, such as folds, and theinterference of surface and body-waves at the edge of the basin can result in significantvariations in site response over short baselines: up to a factor of 2 over 200 m (Hartzellet al. 2010). These structure-related aspects of wave propagation have been the subject ofconsiderable previous studies and continue to be of significant interest in the quantitativeevaluation of seismic hazard (Aki and Larner 1970; Trifunac 1971; Dravinski et al. 1996;Field 1996; Joyner 2000; Rovelli et al. 2001; Spudich and Olsen 2001; Adams et al. 2003;Gvirtzman and Louie 2010; Shani-Kadmiel et al. 2012).
The MW 6.7 Northridge earthquake caused anomalously large ground motions in severalareas in the Los Angeles basin with highly localized damage in Northridge, Sherman Oaksand Santa Monica. For Northridge, the occurrence of large motions and associated damagecan be explained by source processes due to the close proximity to the fault; in ShermanOaks the presence of soft, young sediments at the surface, suggests that near-surface materialrelated amplification played a key role in the damage extent (Graves et al. 1998). However, inSanta Monica, there was little correlation between damage distribution and the presence ofyoung surface sediments: in fact the damage occurred primarily on older, more consolidated(Pleistocene) sediments, with little damage seen on younger, softer (Holocene) materials.This pattern occurred despite the uniform building types and ages throughout the SantaMonica region (Stewart and Ashford 1994).
Gao et al. (1996) suggested, based on measurements of 32 aftershocks, that the enhanceddamage in Santa Monica is explained by focusing of seismic energy by a lens shape structureat a depth of 3–4 km beneath the surface, causing localized amplification of ground motionsby a factor of 5. On the other hand, Alex and Olsen (1998) argue that focusing from thelens-shaped boundary of a 2-D model can only account for 50 % or less of the amplificationin the Santa Monica area. Thus, they conclude that a significant part of the anomalousamplification in Santa Monica area resulted from other effects, such as shallow basin edgeeffects and reverberations in the near-surface low-velocity sediments.
A similar argument was made by Graves et al. (1998) based on a 2-D finite differencesimulation of a well constrained geological model. Their model showed that focusing by a3–4 km deep structure cannot explain the large ground motions observed immediately southof the fault scarp. Alternatively, they suggest that these strong motions were caused by ashallower, 1 km deep, basin-edge structure of a secondary fault. Davis et al. (2000) inverted32 aftershocks of the Northridge Earthquake to show that the damage resulted from thefocusing of seismic waves by three lens shaped features at depths of about 1–3 km.
The diversity of suggested mechanisms to the same end result in the well-studied LosAngeles Basin during the 1994 Northridge earthquake despite the wealth of seismic datafrom the main shock as well as from many aftershocks, illustrates the complexity of theproblem.
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The phenomenon of “seismic waves focusing” was first studied by Aki and Larner(1970) who investigated the effects of irregular interfaces of a layered medium on surfacemotion due to incident plane SH waves. Their pioneering investigation demonstrated thatwhereas flat layer interfaces lead to a relatively simple distribution of surface displace-ment, dominated by resonance effects, irregular interfaces cause complex surface displace-ment dominated by focusing and de-focusing effects which de-emphasize reverberations,i.e., later arriving multiple reverberations are de-emphasized. Trifunac (1971) presenteda 2-D analytical solution for the amplification of mono-chromatic plane SH waves prop-agating through a semi-cylindrical alluvial valley. In a following paper, Wong and Tri-funac (1974) investigated surface motion of a semi-elliptical alluvial valley subjected tomonochromatic incident plane SH waves. Their fundamental work was followed by sev-eral numerical studies that modeled the influence of various geometries and material prop-erties on ground motion. Sánchez-Sesma and Esquivel (1979) computed the scatteringand diffraction of harmonic SH waves by irregularities at the bottom of an alluvial val-ley. Bard and Bouchon (1980a, 1980b) tested the applicability of the Aki–Larner method(1970) and investigated the response of alluvial valleys to incident SH waves. Specif-ically, they studied two separate geometries, one to examine the generation of surfacewaves and the other to examine focusing. Luzón et al. (2004) compared the response ofa sedimentary basin consisting of a homogeneous fill and that of a basin with a velocitygradient.
The advantage of numerical methods over mono-chromatic analytical solutions is theability to model realistic sources, account for attenuation and frequency interference andto differentiate between reverberations dominated amplification, edge-effect or focusing/de-focusing dominated amplification. At the same time the resulting numerical simulations arefrequently difficult to explain in terms of the basic physical processes involved. For instance,by modeling seismic wave propagation in the geologically complex Dead Sea Basin, we havedemonstrated that the simulated ground motions result from a combination of all the physicalprocesses described above (Shani-Kadmiel et al. 2012).
In this study we further develop the line of work described above by using a physicalsource to initiate our numerical simulations and combine analytical methods in order toderive physically-based conclusions about focusing and de-focusing. A simple, bi-materialmodel with controlled initial and boundary conditions is used to outline the geometrical andmaterial constraints that lead to the significant amplification of ground motion by mechanismof geometrical focusing. We present a practical method ensuing from our combined approach,to evaluate the potential for effective geometrical focusing in sedimentary basins and alluvialvalleys for which interface geometry and velocity ratio are known.
2 Study approach
We conduct our study in two manners: (i) an analytical scheme, and (ii) a numerical scheme.The analytical solution uses Snell’s law to predict the location and size of the convergencezone of up-traveling plane-waves for different interface geometries and varying velocityratios. The numerical approach employs a finite difference scheme (E3D, Larsen et al. 2001)and complements the geometrical results of the analytical scheme with amplification ratioprofiles and spectral information.
We distinguish between the different effects, i.e., edge-effect, vertical resonance, hori-zontal resonance, and geometrical focusing using the procedure and visualization techniquesrecently presented in Shani-Kadmiel et al. (2012). Finally, we combine the results of the
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two modeling approaches to constrain geometry and material characteristics of the focusingphenomenon.
3 Analytical solution
We assume a subsurface semi-elliptical interface f (x) = −b√
1 − (x/a)2, between a hostrock with shear-wave velocity VS,r and the basin fill with shear wave velocity VS,b asdescribed in Fig. 1. The vertical axis of the ellipse is denoted b, the horizontal axis denoteda. A semi-ellipse was chosen over other conical sections because the angle of intersectionwith the surface is perpendicular for any a, b combination. In order to keep material relatedeffects to a minimum and concentrate on the geometrical effects our velocity structure waschosen to be bi-material with one uniform velocity for the hosting rock and another uniformvelocity for the basin fill. φi is the angle of ray incidence, φr is the angle of the transmit-ted ray, φn is the angle between the normal to the interface (n̂) at the point of incidenceand the horizontal and φt is the angle between the transmitted ray and the horizontal. Thepoint of convergence for a pair of incident rays crossing the interface at a symmetric dis-tance x from the vertical axis is found by solving for z(x) using Eq. 1 through Eq. 5.We consider hypothetical, above ground points of convergence and assume no refraction or
inci
dent
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basi
n de
pth
incr
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s
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(c) velocity ratio increases
(b)
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Fig. 1 a Schematics and nomenclature of incident and transmitted rays over a semi-elliptical interface betweenhost-rock (dark gray) and basin-fill (light gray). b Schematic cartoon showing pairs of incident rays convergeto different locations, z(x), as a function of point of incidence and basin geometry (increasing basin depth). cSame as (b) for increasing velocity ratio
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reflection at the free surface only for the sake of solution. Our analytical solution assumesvertically propagating plane waves (φu = π/2), and Snell’s law for calculating the refractionangle, φr .
φn(x) = tan−1(− 1
f ′(x)
),−a < x < a, (1)
φi (x) = π − φu − φn(x) (2)
φr (x) = sin−1[(
VS,bVS,r
)sin (φi (x))
](3)
φt (x) = φn(x) + φr (x) (4)
z(x) = −x tan (φt (x)) + f (x),−a < x < a, x �= 0 (5)
We solve for velocity ratios of VR = VS,b/VS,r = 0.25, 0.4, 0.5 and 0.7 and ellipses of width2a = 3 km and depth of 0.4, 1, 2, 3 and 4 km. The eccentricity of these ellipses is
e ={ √
1 − (a/b)2, a ≤ b−√
1 − (b/a)2, a > b= −0.96 to 0.93. (6)
We term the collection of convergence points of many pairs of rays as “convergence zone”.For each set of eccentricity and velocity ratio we calculate the convergence zone. Figure 2presents the results from these calculations for five basin geometries (a = 3 km; b =0.4, 1, 2, 3, and 4 km with eccentricities e = −0.96,−0.75, 0.66, and 0.93, accordingly)and four velocity ratios (VR = 0.25, 0.4, 0.5 and 0.7). The left column presents the height ofthe convergence points z(x) across the basin (−a < x < a). The central column presents thevertical span of the convergence zones, Sz = (zmax − zmin) centered at the average height,zavg = (zmax − zmin)/2, of the convergence zone. The right column presents the angle ofthe transmitted rays, φt (x), (−a < x < 0).
Shallow basins, with negative eccentricity, lead to convergence zones above surface forall velocity ratios. At low VR the height of the convergence zone reaches a maximum ofzmax = 7 km whereas at high VR the height reaches a maximum of 18 km. The transmittedangle, φt , for these shallow basins changes rapidly from moderate angles, of 30◦ to 60◦ atthe edges to near vertical angles, above 70◦ over one eighth of the basin width. The values ofφt at the basin edges becomes larger, approaching vertical, with increasing VR as expectedfrom Snell’s law.
Deeper basins, with eccentricity of 0.66, 0.87 and 0.93, produce convergence zonesdepending on the velocity ratio. For a given eccentricity low velocity ratios will producelower convergence zones and vice versa. For example, for e = 0.66 the minimum height ofthe convergence zone, zmin , is −0.5 and 1.8 km for VR = 0.25 and 0.7, respectively. Forthese deeper basins the transmitted angle, φt , changes gradually over the entire width of thebasin. For example, at VR = 0.4 for e = 0.66, 0.87 the slope of φt is nearly constant. Asexpected, φt at the basin edges becomes larger for higher VR .
The distribution of z(x), can be characterized by its span, Sz , and the average convergencedepth, zavg . The span and average convergence depth of the shallow basins, e = −0.96and −0.75 (Fig. 2b, e), increases with VR and are both well above surface. Deeper basins,e = 0.87 and e = 0.93, converge to a depth below surface, with zavg rising towards surfacewith increasing VR . At eccentricity of 0.66 and VR = 0.4, the interface produces a smallconvergence zone with zavg near the surface (Fig. 2h).
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(d) (e) (f)
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(m) (n) (o)
Fig. 2 Analytical solution results for different Velocity ratios (VR) and eccentricities (e) of a semi-ellipticalmaterial interface. Left column presents the height of convergence, z(x) along the vertical axis for pairs ofvertically incident rays. Center column presents the span, Sz = (zmax − zmin), of the convergence zone. Rightcolumn presents the transmitted angle, φt (x)
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4 Numerical modeling
4.1 Model setup
Our modeling employs the Finite Difference code E3D developed at the Lawrence Liv-ermore National Laboratory (Larsen et al. 2001). E3D is listed by the OECD’s NuclearEnergy Agency (http://www.oecd-nea.org/tools/abstract/detail/ests1300). E3D simulateswave propagation by solving the elastodynamic formulation of the full wave equation ona staggered grid. The solution scheme is 4th-order accurate in space and 2nd-order accuratein time. In this research we employ the software in 2-D mode, which enables us to modelhigher frequencies at lower computational costs. The ability to model higher frequencies isespecially important for studying geometrical effects as their spectral signature spreads tohigh frequencies as will be evident in our results. Wave propagation in our models is initiatedby a buried finite-length fault source with uniform moment. The ruptured plane of the finitesource extends 3.5 km in the up-dip direction, entirely within high velocity rocks centeredbelow the basin as illustrated in Fig. 3. The normal-faulting, double-couple rupture initiatesnear the bottom end of the fault and the rupture front propagates radially from the hypocen-ter along the fault at a constant rupture velocity of 2.8 km/s. All the 2-D elements on thefault plane were given identical moment and a Gaussian source time function with frequencycontent between 0.1 and 10 Hz and corner frequency of 7 Hz. The size of the source (i.e., itsmoment) is not important in this 2-D analysis that allows no energy to dissipate in the thirddimension. Therefore, we only analyze the relative amplification and derive no conclusionsfrom the absolute ground motion.
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Vs=1.0 km/s
Vs=2.54 km/s
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0.41.02.03.04.0
Fig. 3 Basin geometry for the five modeled interfaces and model layout of the basin, fault, rupture and slipdirection used in the numerical solution
We model the following interface geometry: a = 1.5 km and b = 0.4, 1, 2, 3, and4 km (Fig. 3), which corresponds to eccentricity of −0.96,−0.75, 0.66, 0.87 and 0.93. Thevelocity ratios modeled are VR = 0.25, 0.5, 0.4, 0.7. The velocity of the host rock, VS,r ,is kept constant for all simulations while the velocity of the basin fill, VS,b, is changed toreflect the specific VR . Intrinsic attenuation, Q P and QS , was calculated using the empiricalrelations provided by Brocher (2008). Simulation parameters are presented in Table 1.
4.2 Results
In this section we present simulation results for VR = 0.4. The results are summarized andvisualized in Fig. 4 by three panels for each model. From top to bottom they are: (i) horizontalPeak Ground Velocity (PGV) amplification ratio, computed relative to the reference modelof homogeneous rock space (no basin); (ii) time-distance plot of horizontal velocity made upof 360 synthetic seismograms sampled at evenly spaced surface grid cells. The time-distanceimage is overlaid with synthetic seismograms at selected locations; and (iii) frequency-distance plot, presenting the Fourier spectrum of the 360 synthetic seismograms from theprevious frame. The other three sets of results, VR = 0.25, 0.5 and 0.7, are presented in asimilar layout in the supplementary material (S1 and S2 summarize the VR = 0.25 case, S3and S4 summarize the VR = 0. 5 case and S5 and S6 summarize the VR = 0.7 case).
4.3 Description of results
The first three frames in Fig. 4 (a, b, and c) present the results of the reference model. Thefirst arrival of body-waves to the surface occurs just before 4 seconds as seen by the parabolashaped, bright trace in the time-distance plot (Fig. 4b).
Results of the shallow basin, 0.4 km deep are summarized in Fig. 3d–f. The basin exhibitsuniform amplification of 1.5 relative to the reference model, across most of its width (Fig. 4d).The first arrival of direct shear-waves (S), is slightly delayed in the basin relative to thehosting rock and relative to the same location in the reference model (Fig. 4e). A distinctresonance pattern, typical to reverberation of seismic waves in a thin sedimentary layer with
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01234 Reference
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Fig. 4 Numerical simulation results for basin model with velocity ratio VR = 0.4: reference (a–c), 0.4 kmdeep basin (d–f), 1 km deep basin (g–i), 2 km deep basin (j–l), 3 km deep basin (m–o) and 4 km deep basin(p–r). The top frame for each model shows Peak Ground Velocity (PGV) amplification factor along the freesurface. The middle frame shows a time-distance plot of 360 synthetic seismograms evenly sampled at thesurface. Black is positive (right) particle velocity, white is negative (left) particle velocity and gray is nomotion. Selected synthetic seismograms, at their surface position, are plotted on top of the time-distance plotsfor reference. The bottom frame is a frequency-distance plot showing the Fourier transform of each of the 360seismograms in the middle frame. The time-distance plot scale saturates at 0.1 m/s and the frequency-distanceplot scale saturates at 0.5 m/s for clarity
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a strong impedance contrast at its bottom is visible in Fig. 4f. The fundamental frequencyf0 = VS/4h = 0.63 Hz, corresponds to the basin depth (h = 0.4 km) and shear-wavevelocity (VS = 1 km/s). The odd multiples, (2n − 1) f0, overtones, at 1.88 and 3.13 Hz arealso visible.
The 1 km deep basin exhibits higher amplification ratio (Fig. 4g), of 1.8 relative to thereference model, over a narrower region in the central part of the basin with some de-amplification over the basin edges. The first arrival of shear-waves is slightly delayed, at4.5 s (Fig. 4h). The spectral image (Fig. 4i) shows a resonance pattern, whereas the firstmode at 0.25 Hz is not seen, but known from VS/4h, the overtones at 0.75, 1.25, 1.75 Hz areclearly visible.
The ground motions produced by the 2 km deep basin are amplified by a factor of 3over a very narrow strip, less than a kilometer wide at the center of the basin, while groundmotion above the rest of the basin and its surrounding is considerably de-amplified (Fig. 4j.).This localized effect is visualized in the time-distance plot (Fig. 4k and Figure S7 in thesupplementary material) by the large “wiggle” around 5 s at the center of the basin (presentedin greater detail in Fig. 5g). The spectral image (Fig. 4l) shows that the local amplificationin the 2 km deep basin spreads over the entire spectrum, further indicating that this is not asimple resonance effect.
The 3 and 4 km deep basins (Fig. 4m and p, respectively), exhibit basin wide amplificationat varying levels up to a factor of 2. The 3 km deep basin demonstrates ground motionamplification of 1.7 at its center decaying to no amplification at the basin edges. Note thatthis value of amplification is similar to the one observed in the 1 km deep basin (Fig. 4g),where it is found symmetrically off the basin’s center. In the 4 km deep basin, the situationis reversed and the amplification ratios at the basin edges, 2.2, are much more dominant thanthose found at the center, 1.1. The time-distance plots (Fig. 4n, q) show that the S phasewithin the deeper basins are delayed and attenuated relative to the 2 km deep basin. Thespectral images (Fig. 3o, r) show that for the 3 km deep basin the energy is “smeared” overthe entire basin with slightly higher intensity at the center, while in the 4 km deep basin theenergy is concentrated at the basin edges.
4.4 Interpretation of results
The results summarized in Fig. 4 show that with increasing basin depth the following isobserved: arrival of body-waves is delayed and attenuated, and ground motion amplificationoccurs over narrower and limited portions of the basin, namely near its edges or only at itscenter. In spite of the attenuation of delayed body-waves, ground motion amplification at thebasins’ center increases, with the exception of the deepest basin. We therefore further studythe ground motions at the center of the basin’s with greater detail.
Figure 5 presents the synthetic seismogram from the center of each basin between 3 and8 s together with the reference’s seismogram (left column), the Fourier spectrum between0.1 and 10 Hz of the two seismograms (center column) and the spectral amplification ratio ofthe basin’s seismogram computed relative to the reference seismogram. Picks of prominentphases are marked on each seismogram: S is the shear wave phase; R1 is the first arrivalof Rayleigh waves traveling from basin edges, sP is pressure wave phase converted from Sphase at the rock-basin interface and rS denotes arrival of shear waves reflected from therock-basin interface.
The arrivals of S and sP phases are nearly simultaneous in the case of the 0.4 km deepbasin, hence only S phase is marked (Fig. 5a). The initial arrival of body-waves is followed byarrivals of Rayleigh-waves (R1) generated at the basin edges traveling inwards to the center
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b=0.4 km, e=−0.96, =0.4−1.0
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Fig. 5 Synthetic seismograms of horizontal component with phase picks in the basin seismogram (left),Fourier spectra (center) and spectral amplification ratio (right) at the center of the free surface for basins withvelocity ratio VR = 0.4: 0.4 km deep basin (a–c), 1 km deep basin (d–f), 2 km deep basin (g–i), 3 km deepbasin (j–l) and 4 km deep basin (m–o). S is the shear wave phase; R1 is the first arrival of Rayleigh wavestraveling from basin edges, sP is pressure wave converted from S phase at the basin interface and rS is reflectedshear wave. f0 is the fundamental frequency of vertical resonance of SH-waves in the basin fill and fH0 isthe fundamental frequency of surface-waves traveling from the basin edges
of the basin. Later arrivals include, mainly, reflected shear waves (rS) from the basin-bottom,which interfere with each other to cause vertical resonance. The spectral amplification ratio(Fig. 5c) shows a prominent peak, of factor 4, at the fundamental frequency of the soft
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layer, f0 = 0.63 Hz. Overtones of the fundamental frequency are also visible at 1.88 and3.3 Hz.
In the 1 km deep basin the S phase and the sP phase can be differentiated (Fig. 5d). Thetime difference of S phase and R1 phase arrivals is smaller since basin depth increases whileits width remains constant. Spectral amplification ratio shows the fundamental frequency atf0 = 0.25 Hz (Fig. 5f) with amplification factor of 1.7. Amplifications by a factor of 2.7and 4.7 at 0.33 and 0.6 Hz, respectively, are not overtones of the fundamental frequency butare related to structural contributions, specifically the edges of the basin. The frequency forhorizontal resonance of surface waves generated at the basin edges traveling towards eachother can be derived from fH0 = nV/2L , n = 1, 2, 3, . . ., where L is the width of the basin,3 km in this case. Mode 2 is therefore calculated to be 0.33 Hz, marked by fH0 in Fig. 5f, thepeak around 0.6 Hz might be related to the fourth mode which is calculated to be 0.67 Hz.
As described above the ground motion amplification generated by the 2 km deep basin isexceptionally high (Fig. 4g) with all frequencies on the modeled spectrum affected (Fig. 4l).The time-distance plot (Fig. 4k) shows that arrival of shear-waves, the bright parabolic fea-ture between 5 and 6 s, is simultaneous with the arrival of surface-waves, x-shaped linearfeatures. The frequency for horizontal resonance of surface-waves is basin width dependentand hence, remains as computed above, 0.33 Hz. The fundamental frequency of verticalresonance is 0.125 Hz, and its next overtones are 0.375, 0.625 Hz, which are very close to0.33 and 0.67 Hz of the horizontal resonance. We suggest that the amplification caused bythe simultaneous arrival of S and R1 phases is further amplified when horizontal and verti-cal modes of resonance coincide. The spectral amplification ratio (Fig. 5i) exhibits a trendof amplification rising with frequency over the entire range of modeled frequencies until itsharply drops at 7 Hz, the corner frequency of the modeled source.
The simultaneous arrival of body-waves and surface-waves for the 2 km deep basin is bestvisualized with snapshots of the wave propagation solution at selected time steps (Fig. 6). Forclarity, only the horizontal component of particle motion is shown. At 2.1 s the wave-front ofthe S phase is shown in intense colors as it propagates from the finite-length rupture markedas a black line. At 3.5 s the wave-front enters the basin from below; the velocity drop andthe delay of the wave-front in the basin with respect to the host rock is manifested by thechange in curvature of the wave-front. The sP phase surpassing the S phase is visible inlighter blue just above it. At 4.2 s the Rayleigh waves (R1 phase) enter the basin and beginto interfere with the later arriving body-waves from within the basin. At 4.6 s the R1 phaseand the S phase propagate toward the center of the basin. At 4.8 s the R1 phase and S phasearrive simultaneously at the center of the basin. At 5.7 s the surface-waves have crossedeach other and shear-waves reflected from the free surface propagate downward. The highamplitude of vibration within the basin relative to its surrounding is evident from the colorintensity.
Our analytical solution indicates that for basins 3 and 4 km deep with velocity ratioVR = 0.4 the average convergence depth is ∼0.5 and 1.1 km below surface, respectively.Past the convergence zone, body-waves from the right-side crossover those from the left-side,reaching the surface diverged and attenuated. This is visualized in Fig 3n and q: the brightparabolic feature of the S phase is convex rather than concave like in the shallower basins.The R1 phase being basin-width dependent arrives at the same time; ∼4.8 s (Fig. 4j, m). Asthe edges of the basin become steeper and the basin becomes deeper the edge-effect becomesmore dominant since the body-waves traveling to the center of the basin are attenuated over alonger path (Fig. 3o, r). The spectral amplification ratio for the 3 and 4 km deep basins showa peak at 0.33 Hz, a multiple of the surface-waves resonant frequency, with amplificationfactor of 5 (Fig. 4l, o). The 3 km deep basin shows amplification, of an average factor 2, at
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Fig. 6 Wave-field visualization at key time steps for the 2 km deep basin simulation. Red is positive (right)particle motion, blue is negative (left) particle motion; color intensity is proportional to amplitude. Time(seconds) is displayed at the bottom left corner of each snapshot
higher frequencies, whereas the 4 km deep basin shows minor amplifications for frequencieshigher than 0.6 Hz.
5 Discussion
Ground motion amplification due to geometrical focusing of seismic waves may occur above“bowl-shaped” structures. The significance of this effect is determined by the geometry ofthe structure and the velocity ratio across the interface between the sedimentary fill andthe hosting rock. Our combined analytical-numerical approach outlines the geometrical andmaterial constraints resulting in geometrical focusing of seismic waves. We identify three
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separate situations depending on the location of zavg and demonstrate the effect of interfaceeccentricity and velocity ratio on the span, Sz , of the convergence zone.
For zavg above the surface, i.e., the case of the 0.4 km (e = −0.96) and 1 km (e = −0.75)deep basins, ground motion amplification is a result of material related effects (Fig. 4d). Withincreasing basin depth the span decreases and the zavg moves closer to the surface (Fig. 2c,2f). Ground motion amplification in the 1 km deep basin (e = −0.75) occurs over a narrowerpart of the basin with slightly larger values (Fig. 4g). This case is analogous to a smeared,blurry image from a projector focused to a point far behind the screen.
For zavg below surface, i.e., the case of the 3 km (e = 0.87) and 4 km (e = 0.93) deepbasins, the level of ground motion amplification is affected by the attenuation from the pointof convergence to the surface (Figs. 3p, 4m). As the point of convergence is buried deeperbeneath the surface the seismic waves become more diverged reaching the surface at theopposite side from their point of entry. This case is analogous to the mirrored and blurryimage from a projector focused to a point in front of the screen.
When zavg is at or near the surface and the span is sufficiently small, significant groundmotion amplification occurs at the basin center, e.g., the 2 (e = 0.66) km deep basin withVR = 0.4 (Fig. 4j). Out of twenty simulations (results are presented in Figs. 3 and 4 forVR = 0.4 and in supplementary materials Figures S1 through S6 for VR = 0.25, 0.5,and 0.7) only four cases exhibit significant amplification of PGV at the center of the basin(Fig. 7). The basins in these cases, 2 km deep, VR = 0.25; 2 km deep, VR = 0.4; 2 km deep,VR = 0.5; and 4 km (e = 0.93), VR = 0.7, are considered as effectively focusing. The PGVamplification ratio at the center of each basin ranges from 2.5 to 3.2. The spectral signatureof the effectively focusing basin as observed in Figs. 3l and 4i, can also be seen in the spectralimages in Fig. 7 as well (frames c, f, i, and l in Fig. 7). Synthetic seismograms and spectralamplification ratios from the center of each of the effectively focusing basins are presentedin Fig. 8. All cases of effectively focusing basins exhibit ground motion amplification athigh frequencies indicative of geometrical focusing. At low velocity ratios the trend of thespectral amplification is superimposed with a strong horizontal resonance peak related to theinterference of R1 phase. This horizontal resonance peak diminishes at higher velocity ratios.
Our results are in agreement with previous studies, namely Trifunac (1971) and Wongand Trifunac (1974) who presented a 2-D analytical solution for the amplification of mono-chromatic plane SH waves propagating through semi-cylindrical and semi-elliptical alluvialvalleys. In the case of a semi-cylindrical (e = 0) alluvial valley the surface displacementamplification shifts from the center of the basin towards the edges of the basin as the frequencyof the incidence SH wave increases (Trifunac 1971). This shift does not occur in effectivelyfocusing basins as illustrated by the spectral images in Fig. 7 (frames c, f, i, and l).
Wong and Trifunac (1974) show that shallow basins (equivalent to negative eccentricitybasins in our study) exhibit surface displacement amplification that can be readily estimatedby a series of side-by-side 1-D approximations. Deep basins (positive eccentricities) however,show no resemblance in surface displacement amplitude between the 2-D result and the 1-Dapproximation. They discuss two cases in which deep semi-elliptical basins exhibit surfacedisplacement due to geometrical focusing. In the both cases, (i) e = 0.71 and VR = 0.5, and(ii) e = 0.86 and VR = 0.5, surface displacement amplification is up to 4 in the first case and4.75 in the second case and is located at the center of the basin for the studied frequencies.Note that their investigation did not account for intrinsic attenuation and wave interferencesas they consider mono-chromatic waves. Thus, amplification values should not be directlycompared, however the overall focusing phenomenon is evident nonetheless.
Figure 9 contours the average convergences depth, zavg and the span Sz over the parameterspace of interface eccentricity, e, and velocity ratio, VR , of our analytical solution. The four
Fig. 7 Results of numerical simulations for effectively focusing basins: e = 0.66, VR = 0.25 (a–c); e =0.66, VR = 0.4 (d–f); e = 0.66, VR = 0.5 (g–i) and e = 0.93, VR = 0.7 (j–l). Data representation followsthat of Fig. 4
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b=4 km, e=0.93, =0.7−1.0
−0.5
0.0
0.5
1.0
Vel
ocity
, m/s
Time, s
0.001
0.01
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0.1Frequency, Hz
0
2
4
6
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plifi
catio
n ra
tio
0.14 6 8 1 10 1 10Frequency, Hz
(j) (k) (l)
b=2 km, e=0.66, =0.5−1.0
−0.5
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ocity
, m/s
0.001
0.01
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0
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plifi
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(g) (h) (i)
b=2 km, e=0.66, =0.4−1.0
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ocity
, m/s
0.001
0.01
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0
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4
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Am
plifi
catio
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(d) (e) (f)
b=2 km, e=0.66, =0.25−1.0
−0.5
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ocity
, m/s
0.001
0.01
0.1
0
2
4
6
Am
plifi
catio
n ra
tio
(a) (b) (c)
Fig. 8 Synthetic seismograms with phase picks in the basin seismogram (left), Fourier spectra (center)and spectral amplification ratio (right) at the center of the free surface for effectively focusing basins: e =0.66, VR = 0.25 (a–c); e = 0.66, VR = 0.4 (d–f); e = 0.66, VR = 0.5 (g–i) and e = 0.93, VR = 0.7 (j–l)
effectively focusing cases are marked by triangle symbols in Fig. 9. Two additional cases,3 km deep (e = 0.87), VR = 0.5 and 0.7, are marked by open circles. These basins exhibitPGV amplification profile similar to those of the effectively focusing basins; however, theirspectral signature contains a mixture of resonance and geometrical. The rest of the modeledcases are plotted as cross symbols if they exhibited any amplification of ground motion at alland diamond symbols if ground motion was de-amplified basin-wide. In addition, we plot thetwo cases from Wong and Trifunac (1974) as star symbols. We use these cases to estimatethe range of eccentricity and velocity ratio combinations which produce amplification ofground motion by geometrical focusing mechanism. We find that these cases are bound byzavg = ±0.5 km from the surface (marked by the shaded polygon in Fig. 9a) even thoughthe span for these cases varies from 0.5 km to almost 4 km (Fig. 9b).
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−1
km
1 km
2 km
−0.5
km
0.5 km
−0.2
5 km
0.25 km
0 km
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0.30
0.35
0.40
0.45
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0.60
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ocity
rat
io
0.5 0.6 0.7 0.8 0.9
Eccentricity
Average z(x)
1 km
1 km
0.5
km
0.5 km
2 k
m
3km
0.5 0.6 0.7 0.8 0.9
Eccentricity
Span z(x)(a) (b)
Fig. 9 Contours of average convergence depth zavg (left) and span of the convergence zone Sz (right) asfunction of velocity ratio (VR) and interface eccentricity (e). Triangles mark the effectively focusing basins,open circles mark basins which exhibit similar PGV amplification profile with different spectral features,crosses mark basins with amplification of ground motion and diamonds mark basins with ground motionde-amplification. Two cases from Wong and Trifunac (1974) marked as star symbols. Size of symbols isproportional to amplification factor except for diamonds and stars which are fixed size
In our analytical and numerical models we assumed seismic velocity within the basin tobe constant. Although this assumption is not physically justified, particularly for the deeperbasins, it enables us to isolate the structural parameters and compare between the models.Introducing a velocity gradient changes the effectiveness of geometrical focusing and hencethe related ground motion amplification depending on the specifics of the basin. If the velocitygradient introduces a high impedance contrast at the interface, ground motion amplificationis enhanced throughout the basin whereas if the velocity gradually decreases without intro-ducing an impedance contrast at the bottom of the basin, ground motion amplification isshifted from the center to the edges of the basin (Luzón et al. 2004). For simplicity, ouranalytical solution considers upward propagating plane-waves, a reasonable assumption forremote earthquake sources; however, in previous work (Shani-Kadmiel et al. 2012) we havenoticed similar effects for closer sources with non-planar wave-fronts. In our numerical mod-els, wave propagation is initiated by a finite source which represents a physical source. Afinite fault source is preferable over a plane wave as it accounts for rupture size and velocity,propagation and directivity, and spectral content. Note that wave fronts entering the basin arenearly planar (see Fig. 6), thus allowing a favorable comparison between the analytical andnumerical models.
The focal mechanism modeled in this study is that of a normal fault, typically found atthe edges of grabens and deep sedimentary basins. A thrust fault with the same geometryand rupture characteristics will produce the same results but the polarity of the motion willbe inverted. Alluvial valleys and other “bowl-shaped” structures may be subjected to otherpatterns of seismic radiation and seismic waves may enter the basin from its side rather thanfrom the bottom. Wong and Trifunac (1974) show that seismic waves entering the basin fromits side lead to surface displacement amplification close to the velocity interface and exhibitan edge effect pattern rather than geometrical focusing.
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6 Summary and conclusions
The results of our study show that geometrical focusing of seismic-waves by a semi-ellipticalvelocity interface results in significant ground motion amplification. Geometrical focusingoccurs for a very specific combination of interface eccentricities and velocity ratios capable offocusing the seismic waves to a narrow region near the surface. We show that for velocity ratiobetween 0.25 and 0.7 effective geometrical focusing occurs for interfaces with eccentricitylarger than 0.66.
The spectral signature of effective geometrical focusing exhibits ground motion ampli-fication over the entire modeled spectrum (Figs. 6, 7). At overlapping resonance frequen-cies of body-waves (vertical resonance) and surface-waves (horizontal resonance) groundmotion amplification is especially high (Fig. 8). Furthermore, in the time domain, thearrival of these two phases coincides, causing the interference to be constructive. Thus,effective focusing can be viewed as a geometrical effect equating the traveling times ofsurface-waves and body-waves at the center of the basin, “coupling” their amplitudes at apoint.
Sedimentary basins of known geological structure may be tested for geometrical focusing.We suggest the following procedure: (i) identify the strongest seismic reflector; (ii) deter-mine the average shear-wave velocity of the basin fill and the rock-to-basin velocity ratio;(iii) identify concave parts of the velocity interface and determine their eccentricity: If eccen-tricity is negative (shallow basin), there is no risk of effective geometrical focusing and theground motion amplification can be approximated by a 1-D approach. Else, if eccentricityis positive, geometrical focusing may occur; (iv) test for geometrical focusing using Fig. 9.Alternatively one may use Eqs. (1) to (5), which accept general analytical forms of f (x), todetermine the convergence region and span. Based on our numerical analysis the expectedamplification of the PGV at the center of the basin is by a factor of 3, with intrinsic attenuationaccounted for. The suggested procedure will yield a first order approximation which can befurther elaborated using specific numerical modeling to study the spectral signature of groundmotions.
Acknowledgments This research was partially funded by the Ministry of National Infrastructures of theState of Israel, Grant #210-17-001, and by the Geological Survey of Israel as part of a project assessing theinstability factors in the Dead Sea Infrastructure.
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