Research Activities

Two- and Three-Dimensional Effects on Ground Motion

by A. Papageorgiou

This article presents research resulting from NCEER's program in seismic hazards and ground motion. The work described uses a "2.5-D" model developed by the author to model the response of valleys to earthquake excitation. Two different valleys affected by the Loma Prieta earthquake were chosen as case studies. A companion article to appear in a future issue of the Bulletin will detail similar findings from the Northridge earthquake. Comments and questions should be directed to Apostolos Papageorgiou, Rensselaer Polytechnic Institute, at (518) 276-6331; email:

Geologic conditions and topography at or near a site can influence the nature of ground shaking, and their importance on seismic hazard is well known. In particular, sedimentary deposits in the form of sediment-filled valleys or basins often have a pronounced effect on the intensity of strong ground motion. The finite lateral extent of the sedimentary deposit introduces complex effects through the generation of surface waves (referred to as locally generated or valley-induced surface waves) at the edges and resonance in the lateral direction, and tends to increase both the amplitude and duration of ground motion. Therefore, to correctly interpret and/or reliably predict strong ground motion, the response of these geologic structures must be better understood. This can be accomplished by theoretical modeling and by comparing/validating the predictions of these models with recorded data.

The research efforts described in this paper are focused on the response of sedimentary valleys to incident earthquake excitation. Versatile numerical techniques have been developed and implemented to validate models in a number of case studies [e.g., the Caracas Valley (Papageorgiou and Kim, 1991, 1993); the Santa Clara Valley (Pei and Papageorgiou, 1996); the Marina District (Zhang and Papageorgiou, 1995a,b)]. In particular, research focuses on: (a) the amplifying effects that sedimentary basins have on incoming seismic waves; and (b) the excitation of valley-induced surface waves which are generated by diffraction of incident seismic waves at the edges of a basin. These valley-induced (or locally generated) surface waves prolong the duration of motion, induce significant differential motions (strains) and may have detrimental effects on structures such as long period high-rise buildings, base isolated structures and elongated structures such as bridges and pipelines.

The model used in this study consists of an infinitely long viscoelastic inclusion/scatterer of arbitrary but uniform cross-section embedded in a homogeneous (or horizontally layered) viscoelastic half-space. The inclusion/scatterer may represent the sedimentary deposits of an elongated valley or topographic features such as canyons or ridges. The excitation is represented by plane body waves impinging at an oblique angle with respect to the axis of the scatterer, or by plane surface waves incident from any azimuthal direction. Even though the model is two-dimensional, the response is three-dimensional. Such a model is referred to as a "2.5-D" model.

The numerical method that has been developed is a hybrid technique which combines the direct Boundary Integral Equation Method (BIEM) with the Finite Element Method (FEM). The advantage of this technique is that it utilizes the versatility of the FEM to model in detail the scatterer (i.e., valley, ridge or canyon) while the BIEM is used to account analytically for the radiation condition.

As an illustration of the versatility of the numerical method, consider the interaction of an incident plane P-wave with a semicircular canyon embedded in a homogeneous and isotropic elastic half-space. Consecutive "snapshots" of this interaction are shown in figure 1. The scattered wave-field generated by the interaction of the incoming wavefield with the scatterer is evident, and regions of constructive interference may be readily located. Such regions are of concern in the design of underground structures such as subways and underground repositories of hazardous materials.

As a demonstration of the existence of valley-induced surface waves, the response of the Santa Clara Valley to the 1989 Loma Prieta earthquake was analyzed, as recorded by the Gilroy array of instruments which crosses the valley (Pei and Papageorgiou, 1996). The recorded velocity response is shown in figure 2. The presence of surface waves, and in particular Rayleigh waves, is clearly evident. From careful analysis of the particle trajectories of the recorded motions, three Rayleigh modes were identified, including the fundamental. Then, the response of the valley was simulated qualitatively. Figure 3 shows the three velocity components of the synthetic response. The three Rayleigh modes, which were identified in the recorded data, are also present in the synthetic motions.

As a second case study, the response of the Marina District Basin to the 1989 Loma Prieta earthquake was considered. The Marina District was the site of some of the most devastating damage caused by the earthquake, and thus became the focal point of post-earthquake investigations. Unfortunately, no recording instrument existed in the Marina District at the time of the mainshock. For this reason, the simulated mainshock response of a "2.5-D" model of the Marina was compared to the motion recorded at Treasure Island (TRI) which was identified as "the most likely strong motion surrogate for the filled areas of the Marina District" (Hanks and Brady, 1991). The above conjecture was confirmed by simulations and can be seen in figure 4, which compares the synthetic motion of a representative site in the Marina District, with the motions recorded at TRI. It should be pointed out that after a couple of significant acceleration pulses, the TRI site failed due to liquefaction. The phenomenon of liquefaction is not modeled in this analysis.

Based on the results of the simulations, the following may be stated: The non-planar sediment-rock interface of the Marina embayment has a profound effect on weak (small strain) motions, as has already been demonstrated observationally and numerically by Liu et al., (1992) and Graves (1993). However, for strong (large strain) motions, the energy dissipation that takes place in the poorly consolidated sediments (due to the hysteretic stress-strain behavior that soils exhibit) substantially dampens the lateral interferences (i.e., surface wave). Thus, 3-D focusing and lateral interferences, while still present, are not as prominent as in the weak motion (i.e., small strain) excitation case. This is demonstrated in figure 5, where the elastic response of the Marina Basin to a simple pulse (figure 5a) is compared to the response of the Basin to the same pulse but considering damping consistent to the level of strains that the sediments experienced during the 1989 Loma Prieta mainshock.


Graves, R.W., (1993), "Modeling Three-Dimensional Site Response Effects in the Marina District Basin, San Francisco, California," Bulletin of the Seismological Society of America, Vol. 83, pp. 1042-1063.

Hanks, T.C. and Brady, A.G., (1991), "The Loma Prieta Earthquake Ground Motion and Damage in Oakland, Treasure Island, and San Francisco," Bulletin of the Seismological Society of America, Vol. 81, pp. 2019-2047.

Liu, H.P., Warrick, R.E., Westerlund, R.E., Sembera, E.D., and Wennerberg, L., (1992), "Observation of Local Site Effects at a Downhole-and-Surface Station," in The Loma Prieta, California, Earthquake of October 17, 1989 - Marina District, U.S. Geological Survey Professional Paper 1551-F, pp. F51-F74

Papageorgiou, A.S., and Kim, J., (1991), "Study of the Propagation and Amplification of Seismic Waves in Caracas Valley with Reference to the July 29, 1967 Earthquake Response: SH-Waves," Bulletin of the Seismological Society of America, Vol. 81, No. 6, pp. 2214-2233.

Papageorgiou, A.S. and Kim, J., (1993), "Propagation and Amplification of Seismic Waves in 2D Valleys Excited by Obliquely Incident P- and SV-Waves," Earthquake Engineering and Structural Dynamics, Vol. 22, pp. 167-182.

Pei, D, and Papageorgiou, A.S., (1996), "Locally Generated Surface Waves in Santa Clara Valley: Analysis of Observations and Numerical Simulation," Earthquake Engineering and Structural Dynamics (in press).

Zhang, B., and Papageorgiou, A.S., (1995a), "Study of a Linear Two-Dimensional Model of the Marina District, San Francisco, California, using a Hybrid Numerical Technique," Proceedings of the Third International Conference on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, St. Louis, Missouri, April 2-9, 1995 Vol. II, pp. 567-570.

Zhang, B. and Papageorgiou, A.S., (1995b), "Simulation of the Response of the Marina District Basin, San Francisco, California, to the 1989 Loma Prieta Earthquake," Bulletin of the Seismological Society of America, (submitted for publication).

Some of the material reported herein is based upon work supported in whole or in part by the National Science Foundation, the State of New York, the U.S. Department of Transportation, the Federal Emergency Management Agency and other sponsors. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the views of NCEER or its sponsors.

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