International Journal of Geophysics

Volume 2016, Article ID 7131867, 14 pages

http://dx.doi.org/10.1155/2016/7131867

## A Superposition Based Diffraction Technique to Study Site Effects in Earthquake Engineering

Departamento de Ingeniería Civil, Universidad EAFIT, Medellín, Colombia

Received 1 September 2015; Accepted 10 January 2016

Academic Editor: Marek Grad

Copyright © 2016 Juan Gomez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A method to study the response of surface topographies submitted to incident waves is presented. The method is based on the superposition of diffracted sources described in Jaramillo et al. (2013). Since the technique proceeds in the frequency domain in terms of the superposition of incident, reflected, and diffracted waves, it has been termed like a superposition based diffraction approach. The final solution resulting from the superposition approach takes the form of a series of infinite terms, where each term corresponds to diffractions of increasing order and of decreasing amplitude generated by the interactions between the geometric singularities of the scatterer. A detailed, step-by-step algorithm to apply the method is presented with regard to the simple problem of scattering by a V-shaped canyon. In order to show the accuracy of the method we compare our time and frequency domain results with those obtained from a direct Green’s function approach. We show that fast solutions with an error of the order of 6.0% are obtained.

#### 1. Introduction

The presence of surface or subsurface topography has been recognized for many years as an incident factor in earthquake induced ground motions resultant at a site. Classical and well-documented examples can be identified in the large ground accelerations registered at the Pacoima Dam during the 1981 San Fernando, California, earthquake and in the records of Tarzana hill during the 1994 Northridge earthquake [1, 2]. Mathematically, the quantification of topographic effects involves the solution of the elastodynamic wave scattering problem, which in the case of geometries of arbitrary shape gives rise to highly complex responses requiring a numerical solution. Despite the fact that currently existing numerical techniques together with the continuous increase in computational power provide the analyst with an actual capability to conduct large-scale simulations for highly realistic scenarios, closed-form solutions still remain important as validation frameworks, mainly since they derive into important conceptual understanding of the problem of site effects. In this work we introduce a method to determine analytic approximations to the scattering of waves induced by topographic surface irregularities, which in contrast to other analytical treatments available in the literature has the advantage that the solution is progressively built based upon physical arguments giving it great flexibility as a validation tool for numerical implementations or allowing its use as first-hand interpretation of results associated with very complex scenarios.

In the case of scattering of in-plane waves, the number of analytic solutions is limited. For instance, the scattering of incident and waves by a semicircular canyon was only recently found by Lee and Liu [3] using a wave function expansion approach, while in the case of horizontally polarized shear waves there is only a handful of contributions: an important work was conducted by Trifunac [4], who found the frequency domain solution to the scattering of plane waves by a semicircular canyon and a semicircular valley using a separation of variables approach. Similar solutions for the scattering of waves by irregularities of various shapes constructed in terms of wave function expansions have also been developed by a region matching technique in Tsaur and Chang [5]; Tsaur et al. [6]; Tsaur [7]; Liu et al. [8]; Tsaur [9]; Han et al. [10]; Gao et al. [11]; Zhang et al. [12]; Gao and Zhang [13]; Chang et al. [14]; and Tsaur and Hsu [15]. In all of these works the solutions take the form of infinite series or expansions with an infinite number of terms and despite the fact that they correspond to the simplest cases of scattering of scalar waves by strongly idealized geometries, these solutions shed light on conceptual understanding of the problem of site-specific response. One particular limitation of solutions in terms of infinite series is the fact that practical applications are only possible after a thorough numerical analysis, which may be a difficult task due to the nonuniform convergency properties of the expansion functions. These limitations may become important factors when the size of the spatial domain or the frequency content of the analysis is changed.

In a recent contribution, Jaramillo et al. [16] laid down the basis of a superposition analysis technique, where, in contrast to the standard approach behind series solutions, these authors used a partition of the field based upon a physically derived incoming motion representing the geometrical plus the diffracted field. Since the geometrical field can be obtained exactly, using well-known reflection laws, the only approximation to the solution is related to the diffracted field. In the resulting superposition based diffraction (SBD) approach these diffracted terms are derived after representing the topographic irregularity as a superposition of overlapped wedges: each one contributing with a source of diffraction emanating cylindrical waves. The resulting solution is also an infinite series, but by contrast with wave function expansion techniques, here each term corresponds to a diffracted wave of decreasing amplitude whereby truncation is decided based upon accuracy. Depending on the ratio between the wavelength of the incident motion and the characteristic dimensions of the scatterer, the series can be truncated after considering just a few terms which results in a highly economical approach.

In this paper we present a step-by-step description of the proposed SBD algorithm. In the first part we explain our idea of a physically based incoming motion, corresponding to the geometrical (or optical) field, and establish its connection with the classical free-field motion commonly used in earthquake engineering applications. We then review the fundamental solution for the scattering of waves by a generalized wedge as presented in Jaramillo et al. [16]. The paper subsequently describes how to conduct the partition of the computational domain into subdomains according to the regions of existence (or absence) of incident and reflected rays and depending upon the number of diffraction sources introduced by the topographic irregularity. The method is then validated against the solution for a symmetrical V-shaped canyon reported by Tsaur and Chang [5]. We present approximations considering diffracted waves and their interactions with adjacent wedges up to third order. This yields accuracy comparable to the analytic solution. Moreover, in order to show the applicability of the method, we determine the response of a 25° V-shaped canyon using different number of diffraction terms. The SBD results were compared with those obtained by a boundary element method (BEM) computer package. The comparison was conducted in the frequency and in the time domain where we measure the error associated with the different sources of diffraction. Economic solutions, with errors of the order of with respect to the numerical results, are obtained after including just a few terms in the series.

#### 2. Alternative Representation of the Total Response

The basis of the current superposition based diffraction (SBD) approach is the linear character of the problem. This allows the total solution to be written in terms of the addition of different and arbitrary superpositions. One such partition is based upon the usual earthquake engineering definition of free-field motion, where the total response is constructed by the addition of an incident field ; its reflections in a half-space after removing the scatterer ; and a relative or scattered field . Recalling the definition of free-field motion, commonly used in earthquake engineering and given by , allows us to write the total field like

By reasons that will become apparent later, we refer to the free-field term like the artificial incoming motion.

An alternative partition of the field, introduced in Gomez et al. [17] and constituting the basis of our method, is now explained with reference to Figure 1 which schematically describes a scattering problem where the domain (a) has been partitioned into the scatterer (c) and its supporting half-space (b), respectively.