Scattering of elastic waves by dipping layers of arbitrary shape embedded within an elastic half-space is investigated for a plane strain model by using a boundary method. Unknown scattered waves are expressed in the frequency domain in terms of wave functions which satisfy the equations of motion and appropriate radiation conditions at infinity. The steady state displacement field is evaluated throughout the elastic medium for different incident waves so that the continuity conditions along the interfaces between the layers and the traction-free conditions along the surface of the half-space are satisfied in the least-squares sense. Transient response is constructed from the steady state one through the Fourier synthesis.
The results presented show that scattering of waves by dipping layers may cause locally very large amplification of surface ground motion. This amplification depends upon the type and frequency of the incident wave, impedance contrast between the layers, component of displacement which is being observed, location of the observation station and the geometry of the subsurface irregularity. These results are in agreement with recent experimental observations.
INTRODUCTiON
This paper is an extension of the study by Eshraghi and Dravinski' in which scattering af plane SH waves by dipping layers of arbitrary shape was considered. The present paper extends the analysis to the plane strain model of the problem. A detailed literature review of the problem can be found in the companion paper by Eshraghi and Dravinski' and it will not be repeated here.
As discussed in the paper dealing with the antiplane strain model,' calculation of strong ground motion due to subsurface irregularities may require a great amount of computation time. Consequently, this paper considers a method which appears to be computationally very effective in comparison to the standard numerical techniques such as finite elements and finite difference. In this method, the unknown scattered wave field is expressed in terms of the wave functions which satisfy the equations of motion and appropriate radiation conditions at infinity. However, these functions do not satisfy the stress-free boundary conditions along the surface of the half-space. This idea originates in works of Herrera and Sabina' and Herrera3 and it has been successfully applied to problems of scattering of elastic waves by Sanchez-Sesma et al. 495 and Dravinski et ~1 . ~
STATEMENT O F THE PROBLEM
The geometry of the problem is depicted by Figure of the paper by Eshraghi and Dravinski'. The problem model consists of an elastic half-space with a finite number of elastic dipping layers of arbitrary shape. The layer interfaces are considered to be smooth, with no sharp corners. The conventions are the same as in the paper dealing with the antiplane strain model.' Subscript j corresponds to either layer domains D j ( j = 0, 1, . . . , R ) or to the interfaces C j ( j = 1, . . . , R). Domain Do denotes the half-space layer while C, denotes the interface between the half-space and the first layer, etc. Summation over repeated indices is understood. IJnderlined indices indicate that the summation is being suppressed.