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ONE- VS TWO- OR THREE-DIMENSIONAL EFFECTS IN SEDIMENTARY
VALLEYS
Jacobo BIELAK1, Yoshiaki HISADA2, Hesheng BAO3, Jifeng XU4 And Omar GHATTAS5SUMMARY
This study of the effects of local geological conditions on seismic ground motion uses 1D
amplification as a reference point and examines, via simple theoretical and more realistic
numerical examples and observations, how 2D and 3D conditions differ from 1D estimations.
Because 1D simulations cannot model basin and edge effects, 1D response tends, in general, toexhibit lower peaks and be of shorter duration than 2D and 3D results. On the other hand, due todestructive interference of different types of waves, there are sites where the response can be muchsmaller than predicted by 1D models.
INTRODUCTION
It has long been known that local geological conditions can strongly influence ground motion during earthquakes(Matsuzawa, 1926). The importance of these effects became dramatically apparent during the 1985 Michoacánearthquake, which caused widespread damage in Mexico City, at a distance of 400 km from the epicenter.
Damage was extensive to structures located within the lakebed region, but was minimal or nonexistent outside
the soft soil deposits Since then, the 1988 Spitak, 1989 Loma Prieta, 1994 Northridge, and 1995 HyogokenNambu (Kobe) earthquakes have underscored the importance of the effects of site conditions on groundresponse. In particular, the toll from the Kobe earthquake was huge, in excess of 5000 dead and 200US billion
dollars in direct material losses even though its magnitude (6.8) was of the same order as that of the Northridge
earthquake, which caused severe, but far less damage. Most of the damage in the city of Kobe occurred within
the so-called “disaster belt,” a narrow strip 1 km wide and 20 km long about 1 km away from the Osaka basin.
The combined effects of the source (the forward directivity effects, e.g., Wald (1995)) and the site (the basinedge effects, e.g., Kawase (1966)) probably caused the large ground motion within the belt.
Local site effects are generally taken into consideration in seismic provisions of building codes by means of a
specified amplification factor applied to the spectral acceleration coefficient. This amplification factor depends
either on the type of surficial soil at the site (e.g., NEHRP, 1997) or on the location of the site (e.g., Mexico City
Building Code, Reglamento, 1993). In both cases, the value of this amplification factor is derived fromobservations and from one-dimensional (1D) amplification studies. Evidence from recent earthquakes, however,indicates that 2D and 3D site effects on ground motion are often important. It remains controversial whether inpractice it is sufficient to consider 1D models or whether 2D or 3D models are necessary to evaluate realisticallythe site effects, since 1D models neglect completely the surface waves.
In this paper we summarize key results of 1D dynamic amplification of soil deposits and review, using
observations and results from simulations, key aspects of the 2D and 3D site effects on soil response in
sedimentary valleys. We restrict our discussion to the linear range of response.

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Department of Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh PA, USA; jbielak@cmu.edu
Kogakuin University, Department of Architecture, Tokyo,Japan
Algor, Inc., Pittsburgh, PA, USA
Lam Research, Inc., Fremont, CA, USA
Department of Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh, PA, USAONE-DIMENSIONAL EFFECTS
One argument advanced by engineers to justify the use of vertically incident waves as the excitation is that theincoming body waves tend to become vertical as they travel upwards, as a consequence of Snell’s law. This is
illustrated in Fig. 1, which shows the particle displacements on a vertical cross-section of the San Fernando

Figure 1. (a) Instantaneous displacement pattern in San Fernando Valley (from Quake animation,www.cs.cmu.edu/~quake; (b) Dimensionality of problems used for studies of site effects (Hisada and Yamamoto, 1996)

Valley due to a double couple applied at a point 11 km deep at a dip angle of 44o, representing an aftershock of
the 1994 Northridge earthquake. Naturally, the excitation for an extended fault is far more complex.
To fix ideas concerning 1D amplification in layered systems it is useful to consider the simple case of a single
elastic layer of thickness H underlain by an elastic halfspace, as shown on the top left corner diagram of Fig. 1b.
This case has been studied by many investigators (e.g., Kanai et al (1959), Herrera and Rosenblueth (1965),
Roesset, (1970)). We consider a vertically incident steady-state harmonic SH-wave of unit amplitude andfrequency ω. The amplitude of the displacement, u, at the free-field surface at the top of the layer is then given
by:
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(1)
u w = cos2 (ωH Vs )+α 2 sin 2 (ωH Vs ) 2
In this equation, w is the amplitude of the displacement that would occur at the top of the halfspace if the layer
were not present. Hence, the ratio |u|/|w| then represents the amplification of a layer of density ρs, and shear wave
velocity Vs with respect to that of the halfspace alone. ρr and Vr are the corresponding values for the halfspace,and α=ρsVs/ρrVr is the impedance ratio. From (1) it follows that |u|/|w| attains maximum and minimum values atfrequencies ωmH/Vs = (2m-1)π/2, mπ/2, m=1,2,…, respectively. Notice that the resonant frequencies depend
only on the layer thickness and shear wave velocity, independently of the properties of the halfspace. The sameis true for the frequencies for which the response is minimum. The corresponding values of the amplification
ratios are 1/α and unity. These values decrease if damping is incorporated into the soil model (see, e.g., Roesset,1970). For multiple layers the response is qualitatively similar, except that the separation between resonant
frequencies becomes variable and the corresponding peaks take different values.

[]

TWO-DIMENSIONAL EFFECTS
To examine the effect that lateral confinement has on the resonant frequencies of a layer, we consider initiallythe extreme case of a box of rectangular cross section of width L, height H, and infinite length. The box, which isfixed at the bottom and lateral sides, and free at the top, is filled with a homogeneous material of density ρs andshear wave velocity Vs. Bard and Bouchon (1985) and Bielak et al (1999) considered this example to help
explain the behavior of 2D valleys under SH excitation. The natural frequencies of this valley, normalized withrespect to the fundamental frequency of the single layer, are given by:

[

]

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ω mn ω 1 = (2m − 1) + n (2 H L ); m, n = 1,2,...
(2)This expression shows that the valley has a double set of natural frequencies. The first term on the rhs
corresponds precisely to the mth resonant frequency of the infinite layer without lateral confinement. The second
term reflects the influence of the lateral walls. For each natural frequency of the flat layer there is an infinite set

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of frequencies stemming from the lateral confinement. Associated with each frequency ωmn there is a naturalmode, umn, given by:(3)
u mn = sin (2 m − 1)πy 2 L sin (nπx L ); m, n = 1, 2,...
The origin of the coordinate system is at the lower left corner of the valley, the horizontal x-axis along the width,
and the vertical y-axis toward the free surface. The first factor on the rhs gives the modal shapes of the flat layer
while the second is the modification due to the finite lateral length of the valley. The modes have nodes along thefree surface, where the displacement vanishes. One way to interpret these nodes is to regard the motion in thevalley as consisting of waves. There are body waves traveling upwards from the bottom of the valley that get
reflected at the free surface. In addition, surface waves, which are generated at the two lateral edges of the valley
travel from one edge to the other. For large or open basins these surface waves dissipate due to attenuation
before they reach the opposite edge. In either case, large response occurs when upward and lateral waves at apoint have the same sign. Cancellation occurs at the nodes when the two waves have opposite signs. These areforms of constructive and destructive interference that can be observed in more complex situations. For instance,
Bielak et al (1999) simulated the ground motion in a small, closed, valley in Kirovakan, located near the
epicenter of the 1985 Spitak earthquake in Armenia to try to explain observed damage for which 1D analyses
had substantially uderpredicted the ground surface motion (Yegian et al, 1994). Figure 2a shows the 1D and 2D
amplification ratios (AR) at a point in the middle of the valley as a function of the frequency of excitation, with[]

Figure 2. (a) 1D and 2D amplification ratiosatsite F(see insert) due to steady-state harmonic vertically incident SH-wave; (b)Maximum free surface 1Dand 2D syntehtic accelerations across a valley in Kirovakan during Spitak earthq. Bielak et al (1999).

respect to the free-field response. For the frequency range considered, the AR exhibits three resonancefrequencies, at somewhat shorter intervals than for a single layer, and the peak values no longer have a constantvalue as for the single layer. The 2D amplification ratio also exhibits resonant behavior in the vicinity of the 1D
resonant frequencies, but the corresponding 2D frequencies are slightly higher than those for the 1D results, due
to the lateral confinement of the valley. The values of the respective peaks, however, are considerably larger forthe 2D case. In addition to the essentially 1D resonant frequencies, the 2D valley exhibits resonant behavior atother frequencies, similarly to the simple box. The amplification ratio oscillates rapidly with frequency, reaching
peak values that greatly exceed the 1D values. Interestingly, for certain frequencies the 2D amplification ratio ismuch smaller than unity, denoting in effect a strong deamplificaton of seismic waves akin to that observed forthe box. In addition to steady-state excitation, the 1D transient response of the valley was calculated for atransient incoming wave based on a reference acceleration recorded during the 1994 Spitak earthquake. Figure
2b shows the spatial variation of the peak ground acceleration along the entire valley, corresponding to both 2D
and 1D simulations. The peak response is highly oscillatory within the softer portion of the valley, exhibiting
marked edge effects at the two margins of the valley. At the right edge both 1D and 2D peak accelerations arelarge, as a consequence of the large impedance contrast between the softest soil near the free surface and thesurrounding rock. At the left margin, the 2D response is markedly higher than for 1D, the difference being due towave diffraction and surface waves.
A similar but more pronounced edge effect was obtained by Adams et al (1999) in a simulation of the responsein the Lower-Hutt Valley in New Zealand, due to a Ricker pulse with a central frequency that coincides with thefundamental 1D resonant frequency of the deeper part of the valley. The 1D and 2D transient response isdepicted in Fig. 3 as a set of surface ground velocity seismograms along the entire length of the valley. Notice

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Figure 3. (a) 1D and (b) 2D synthetic velocity acrossa model of Lower Hut, NZ; vertically incident SH-wave with a Ricker pulse
excitationwith central frequency of 2 Hz (Adams et al, 1999; Adams, personal comm. 1999).

the clear trace of the surface wave in the 2D simulation and the large 2D amplification near the left edge causedby the steep discontinuity between the valley and the surrounding rock. Although this valley is closed, surface
waves do not reach the opposite edges during the strong motion phase due to the large extent of the valley.
Figure 4 provides another illustration of 2D site effects on ground motion, for an open basin. It shows the 2D

Figure 4. (a) Map of Yokohama indicating location of instrumented sites; (b) observed and synthetic spectral 1D ratiosat STUT and
SHIR stations due to an actual earthquake (Sato et al, 1996)

spectral, or amplification, ratios at two locations in the Sendai vicinity in Japan obtained by Satoh et al (1995)
from records registered during an actual earthquake. The location SHIR and TSUT are identified in Fig. 4a. Also
shown are results of 1D simulations by these authors using soil properties determined through an inversion
process. The similarity between the 1D simulation results and the direct observations at SHIR is quite striking.
This indicates that a 1D analysis is sufficient for determining site effects at SHIR, which is located in the central
portion of the basin. By contrast, at STUT, which is located closer to the basin edge, 2D effects similar to those
calculated for Kirovakan are observed.
THREE-DIMENSIONAL EFFECTS
The occurrence of recent earthquakes in highly populated regions has spurred the placement of instruments forrecording strong ground motion in basins. In addition, the advent of powerful workstations and parallel
computers and the desire to gain a better understanding of the spatial and temporal distribution of ground motion
in basins has enabled investigators to conduct 3D simulations of ground motion using realistic models (e.g.,Frankel, 1993; Graves, 1996; Olsen and Archuleta, 1996; Bao et al, 1998; Pitarka and Irikura, 1998). In thissection we present several simulation results and observations to highlight some of the 3D effects of surface
geology on ground response to earthquakes.
We start with a simple model shown in Fig. 5a of an idealized closed basin, first used by Sánchez-Sesma andLuzón (1995). The basin consists of soil whose shear wave velocity is one half that of the exterior medium and is

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Figure 5. (a) Plan view of idealized (Moon) basin; (b) 1D and 3D synthetic horiz. displacements (Bao, 1998).

subjected to a Ricker pulse travelling in the form of an incident SV-wave with a central frequency equal to the
1D fundamental frequency of the basin at the deepest point (Bao, 1998). Figure 5b shows synthetics of thehorizontal velocity along the y-axis for a vertically incident wave with particle motion polarized along the x-axis.
Also shown is the corresponding 1D simulation. There are two main differences between the two sets of results.
First, the larger peak of the 3D response and second, its significantly extended duration. Figure 6a shows the

Figure 6. (a) Synthetic seismoscope trace of horizontaldisplacement at several locations within the Moon basin and its vicinity due to
Ricker pulse excitation; (b) Distribution of maximum horiz. velocities (Bao, 1998).

particle orbits on the horizontal plane at a number of points thorughout the basin for different azimuths of theincoming wave. This angle, denoted as ϕ is measured from the x-axis. The incident wave makes an angle of 30owith the vertical. While the particle motion generally tends to follow the direction of motion of the incident
wave, there is a strong 3D effect characterized by the deviation from the direction of incidence. The distributionof the maximum amplitude of the horizontal velocity with respect to that in the free field is shown in Fig. 6b.
The maximum amplification is of the order of 3 for all incident angles. The spatial distribution of the response,
however, varies significantly with the angle of incidence. In particular, the surface waves are greatly amplified at
the opposite end from which the waves arrive.
Figure 7 shows displacements observed and simulated at two sites in Kobe during an aftershock of the 1995

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Figure 7. (a) Band-passed displacements (0.9-5.0 s) observed and simulated for a 1995 Kobe earthq. aftershock at sites KBU and
RKI (see Fig. 10a) (Hisada et al, 1998).

Kobe earthquake. The location of these sites can be seen in Fig. 10a. Due to the large extent of Osaka Bay theground motion in Kobe City essentially corresponds to that in an open basin. All the records are band-passed
filtered for 0.9-5 seconds (Hisada et al, 1998). For the simulations denoted as GRN we used the Green-function
method of Hisada (1995), assuming a 3D wavefield in the 1D flat-layered structure below each site. The
simulations labeled FEM are fully 3D. They use a 3D model of the Kobe region and the FEM to calculate theground motion (Bao et al, 1998). Records denoted as OBS are observations. For KBU, which is located justoutside the basin, the GRN and FEM show an excellent agreement with each other and with the actual records.
At RKI, a site well inside the basin, the agreement between the 3D simulations and the observations is also
excellent. By contrast, the flat-layer simulations exhibit large discrepancies with the observations. The reason forthis is that the flat-layered system cannot model the surface waves generated at the edge of the basin, and thus
completely misses the strong phase that arrives at about 12 s, which is stronger than the direct wave.

Site Response

To illustrate how ground motion can vary rapidly over very short distances during actual earthquakes, Fig. 8

Frequency (Hz)Figure 8. (a) Location of sites in SFV instrumentedfor studying spatial variation of ground motion of 1994 Northridge earthq.
aftershock; (b) spectral ratios at indicated locations derived from recorded seismograms (Hartzell et al, 1996).

shows seismograms of ground motion recorded by Hartzell et al (1996) in two different areas of the San
Fernando Valley during aftershocks of the Northridge earthquake. Hartzell et al obtained records at a large
number of sites, as shown in Fig. 8a. The Sherman Oaks area, which suffered extensive damage during the main
shock is located in the southern part of the valley. Figure 8 shows the spectral ratios at two of these locations,which are separated by less than 100 m. This ratios differ by factors of the order of 3, even at a frequency as low
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as 1 Hz. Similar results are observed for the sites ia1 and ia5, located at Interstate 5, just at the northern edge ofthe valley, nominally on rock.
Graves (1999) performed simulations of the main shock of the 1995 Kobe earthquake over an extended region
that covers both Kobe and Osaka, as shown on Fig. 9a. Figure 9b shows comparisons between 3D and flat

Figure 9. (a) Kobe-Osaka region showing instrumented locations; (b) observed and simulated velocities at two sites during 1995
Kobe earthquake (Graves, 1998).

layered simulations and observations at two sites. The 3D simulations capture the main features of the actual
ground motion, albeit with significant quantitative differences. The flat-layered simulations, on the other hand,exhibit much larger differences and completely miss the strong surface waves generated at the northwestern edgeof the basin parallel to the causative fault.
Matsushima and Kawase (1999) conducted simulations of Kobe earthquake as well, using a refined model of thegeological structure over a limited region comprising the Kobe City surroundings and a detailed model of thesource that contains four asperities. Results for the sites FKA, KBU, and RKI (which can be seen in Fig. 10a) are
shown in Fig. 10. The agreement between the simulations and the observations is quite close at all three sites.

Figure 10 (a) Kobe region showing instrumented locations; (b) observed and simulated velocities at three sites during 1995 Kobe
earthquake (Matsushima and Kawase, 1998).

This suggests that model-based simulation show promise as a practical tool for performing the first, critical stepin the earthquake resistant design process, which is to assess the ground motion to which a structure will be
exposed during its lifetime.

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CONCLUDING REMARKSIn this study of the effects of local geological conditions on seismic ground motion, we have used 1D
amplification as a reference point and examined, via simple theoretical and more realistic numerical examples
and observations, how 2D and 3D conditions differ from 1D estimations. Because 1D simulations cannot modelbasin and edge effects, 1D response tends, in general, to exhibit lower peaks and be of shorter duration than thecorresponding 2D or 3D response. On the other hand, due to destructive interference of different types of waves,there are sites where the response can be much smaller than predicted by 1D models. In fact, one of the mostimportant differences between 1D modeling and 2D, 3D modeling and observations is that while actual groundmotion can vary rapidly over short distances even with little or no change in local soil properties, rapid changes
in 1D modeling can occur only if the soil properties vary abruptly in the lateral directions. While not yet
straightforward, it appears possilbe to develop seismic design provisions that build on existing ones based on 1D
amplification studies for taking local site conditions into consideration. This could be done, for instance, byintroducing factors that would incorporate the basin and edge effects as functions of the soil properties
underneath the site and of its location with respect to the basin edge. Results of 1D, 2D, and 3D simulations andobservations suggest that these factors, however, may vary with frequency.
One final note is in order. Most of the 2D and 3D simulations to date are based on linear soil material models.
Energy dissipated due to inelastic behavior, however, causes a reduction in the amplitude of the surface waves
that varies exponentially with distance from the edge. Thus, it can be expected that for strong earthquakes edge
effects on ground motion are not as large as predicted by purely linear models, especially away from the valleyedges. Basin effects from overall local geological structure, on the other hand, can be expected to remain present
throughout the basin.
ACKNOWLEDGMENT
We would like to thank Prof. Keiiti Aki for his continuous encouragement and advice, and Prof. Francisco J.
Sánchez-Sesma for many useful discussions. Daniel Bielak helped prepare the manuscript for publication This
research was supported by the U.S. National Science Foundation’s Grand Challenge program, under grant CMS9318163. The cognizant NSF program director is Dr. Clifford J. Astill. Computing services on the Pittsburgh
Supercomputing Center’s Cray T3D were provided under grant BCS-960001P. We are grateful for this support.
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