Article A note on elastic noise source localization Abdul Wahab and Rab Nawaz Abstract The problem of reconstructing the spatial support of ambient noise sources from elastic wavefield boundary measure- ments using cross-correlation techniques is dealt with. It is demystified that the cross-correlation-based standard source localization functional in elastic media does not provide optimal refocusing due to different pressure and shear wave speeds. Then, a weighted functional is proposed to rectify the coupling artifacts. A numerical experiment is presented to substantiate the appositeness of the proposed functional. Keywords Noise source localization, elasticity imaging, inverse source problem 1. Introduction We consider the problem of reconstructing the spatial support of ambient noise sources from boundary wave- field measurements in an isotropic homogeneous elastic medium in two or three dimensions using cross-correla- tion techniques (Borcea et al., 2010; Garnier and Papanicolaou, 2012; Garnier et al., 2013; Hoop et al., 2013). The main application envisaged for the present work is so-called passive elastography where the aim is to identify the muscle noise sources in an isotropic elastic medium (Gennisson et al., 2003; Sabra et al., 2007; Archer and Sabra, 2010; Carmona, 2011; Ammari et al., 2014). Another potential applica- tion is the localization of the Earth’s background noise source distribution, which contains significant informa- tion about the regional geology, time-dependent crustal changes and earthquakes (Asghar et al., 1998; Garnier and Papanicolaou, 2009; Kader, 2011; Hoop et al., 2013; Nawaz and Lawrie, 2013). Nayfeh (1995), Chen et al. (2008), Kuske (2010), Shen et al. (2013), Afzal et al. (2014) and Nawaz et al. (2014) detail other potential applications and techniques associated with the present work. The problem of ambient noise source localization in acoustic media (both attenuating and nonattenuating) has been considered by Ammari et al. (2012) wherein cross-correlation-based imaging functionals were estab- lished. In this note, we extend the same approach to elastic media. We first consider the elastic counterpart of the source localization functional used by Ammari et al. (2012). Unfortunately, it mixes the irrotational and solenoidal components of the source due to different pressure and shear wave speeds. Nevertheless, we present a new weighted functional, based on a Helmholtz decomposition for the Green function (initially proposed by Ammari et al. (2013)), taking into account the different wave speeds for pres- sure and shear waves. 2. Mathematical formulation Let   R d , d ¼ 2,3, be an open bounded domain, occu- pied by a homogeneous isotropic elastic material, with Lipschitz boundary @. Consider the linear elastic wave equation in R d , that is @ 2 u @t 2 ðx, tÞL l, uðx, tÞ¼ nðx, tÞ, t 2 R, uðx, tÞ¼ @u @t ðx, tÞ¼ 0, x 2 R d , t  0 8 > < > : ð1Þ for all x 2 R d where L l, u ¼ u þðl þ Þrðr  uÞ ð2Þ Department of Mathematics, COMSATS Institute of Information Technology, Pakistan Corresponding author: Rab Nawaz, Department of Mathematics, COMSATS Institute of Information Technology, 47040, Wah Cantt., Pakistan. Email: rabnawaz@ciitwah.edu.pk Received: 10 March 2014; accepted: 8 July 2014 Journal of Vibration and Control 2016, Vol. 22(7) 1889–1894 ! The Author(s) 2014 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1077546314546511 jvc.sagepub.com at Korea Advanced Institute of Science and Technology (KAIST) on April 13, 2016 jvc.sagepub.com Downloaded from