Elastic least-squares reverse time migration using the energy norm Daniel Rocha 1 and Paul Sava 1 ABSTRACT Incorporating anisotropy and elasticity into least-squares migration is an important step toward recovering accurate amplitudes in seismic imaging. An efficient way to extract reflectivity information from anisotropic elastic wavefields exploits properties of the energy norm. We derive linearized modeling and migration operators based on the energy norm to perform anisotropic least-squares reverse time migration (LSRTM) describing subsurface reflectivity and correctly predicting observed data without costly decomposition of wave modes. Imaging operators based on the energy norm have no polarity reversal at normal incidence and remove backscattering artifacts caused by sharp interfaces in the earth model, thus accelerating convergence and generating images of higher quality when compared with images pro- duced by conventional methods. With synthetic and field data experiments, we find that our elastic LSRTM method generates high-quality images that predict the data for arbi- trary anisotropy, without the complexity of wave-mode de- composition and with a high convergence rate. INTRODUCTION The search for more reliable seismic images and additional subsur- face information, such as fracture distribution, drives advances in seismic acquisition, such as larger offsets, wider azimuths, and multi- component recording. All of these advances facilitate incorporating anisotropy and elasticity into wavefield extrapolation and reverse time migration (RTM), which is the state-of-art wavefield imaging algorithm suitable for complex geologic structures (Baysal et al., 1983; Lailly, 1983; McMechan, 1983; Levin, 1984; Chang and McMechan, 1987; Hokstad et al., 1998; Farmer et al., 2009; Zhang and Sun, 2009). Although seismic acquisition improves with such advances, it always involves practical limitations, such as finite and irregular data sampling, which negatively impact wavefield im- aging methods. Even if anisotropy and elasticity are assumed in wavefield migration, such limitations still exist. Consequently, this type of migration often leads to images with poor resolution and unbalanced illumination due to such practical acquisition constraints, even though image amplitudes are more reliable when compared with acoustic and/or isotropic imaging (Lu et al., 2009; Phadke and Dhu- bia, 2012; Du et al., 2014; Hobro et al., 2014). A common solution to these limitations is the implementation of least-squares migration (LSM), which iteratively attenuates artifacts caused by truncated acquisition and provides high-quality images that best predict observed data at receiver locations in a least- squares sense (Chavent and Plessix, 1999; Nemeth et al., 1999; Kuehl and Sacchi, 2003; Aoki and Schuster, 2009; Dong et al., 2012; Yao and Jakubowicz, 2012). In particular, if the two-way wave equation is used for extrapolation, the method is called least-squares RTM (LSRTM) (Dai et al., 2010; Dai and Schuster, 2012; Dong et al., 2012; Yao and Jakubowicz, 2012). However, to overcome these issues from acquisition and to exploit the advan- tages of more realistic wave extrapolation, some authors propose LSRTM that accounts for multiparameter earth models that can ei- ther incorporate solely anisotropy (Huang et al., 2016), elastic (Alves and Biondi, 2016; Feng and Schuster, 2016; Xu et al., 2016; Duan et al., 2017; Ren et al., 2017), or viscosity effects (Dutta and Schuster, 2014; Sun et al., 2015). For instance, the viscoacous- tic and pseudoacoustic implementations define earth reflectivity in terms of contrast from a single model parameter (Dutta and Schus- ter, 2014; Huang et al., 2016) or in terms of a scalar image based on conventional crosscorrelation between wavefields (Sun et al., 2015). Alternatively, elastic LSRTM implementations in isotropic media provide multiple images that are defined in terms of cross- correlation between decomposed wave modes (Alves and Biondi, 2016; Feng and Schuster, 2016; Xu et al., 2016; Duan et al., 2017). However, wave-mode decomposition in anisotropic media is costly and not as straightforward as in isotropic media; therefore, aniso- tropic wave-mode decomposition remains a subject of ongoing re- search (Yan and Sava, 2009, 2011; Zhang and McMechan, 2010; Cheng and Fomel, 2014; Sripanich et al., 2015; Wang et al., 2016). Manuscript received by the Editor 20 July 2017; revised manuscript received 30 October 2017; published ahead of production 14 January 2018; published online 13 April 2018. 1 Colorado School of Mines, Center for Wave Phenomena, Golden, Colorado, USA. E-mail: drocha@mines.edu; psava@mines.edu. © 2018 Society of Exploration Geophysicists. All rights reserved. S237 GEOPHYSICS, VOL. 83, NO. 3 (MAY-JUNE 2018); P. S237–S248, 9 FIGS. 10.1190/GEO2017-0465.1 Downloaded 09/20/18 to 138.67.129.34. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/