Isotropic elastic wavefield imaging using the energy norm

From the elastic-wave equation and the energy conservation principle, we have derived an energy norm that is applicable to imaging with elastic wavefields. Extending the concept of the norm to an inner product enables us to compare two related wavefields. For example, the inner product of source and receiver wavefields at each spatial location leads to an imaging condition. This new imaging condition outputs a single image representing the total reflection energy, and it contains individual terms related to the kinetic and potential energy (strain energy) from both extrapolated wavefields. An advantage of the proposed imaging condition compared with alternatives is that it does not suffer from polarity reversal at normal incidence, as do conventional images obtained using converted waves. Our imaging condition also accounted for the directionality of the wavefields in space and time. Based on this information, we have modified the imaging condition for attenuation of backscattering artifacts in elastic reverse time migration images. We performed numerical experiments that revealed the improved quality of the energy images compared with their conventional counterparts and the effectiveness of the imaging condition in attenuating backscattering artifacts even in media characterized by high spatial variability. INTRODUCTION Seismic wavefield imaging is usually implemented using the acoustic-wave equation because of the inaccurate assumption that only P-waves propagate in the subsurface. The search for more authentic images and subsurface information such as fracture distribution drives the development of wavefield imaging using the elastic-wave equation. Multicomponent seismic recording and improved computer resources have made elastic wavefield imaging possible (Denli and Huang, 2008; Yan and Sava, 2011; Yan and Xie, 2012; Chen and Huang, 2014; Duan and Sava, 2014a, 2014b). For acoustic and elastic cases, wave-equation migration consists of two steps: (1) wavefield extrapolation in the subsurface using data recorded at the surface or the known source function and (2) the application of an imaging condition for the purpose of extracting the earth’s reflectivity from wavefields (Claerbout, 1971; Dellinger and Etgen, 1990; Yan and Sava, 2009). If a two-way elastic-wave equation is used in the wavefield extrapolation step, followed by an imaging condition representing zero-lag crosscorrelation between the wavefields, the imaging procedure is called elastic reverse time migration (RTM) (Chang and McMechan, 1987; Hokstad et al., 1998). In recent years, many elastic imaging conditions have been proposed by exploiting the multicomponent aspect of the elastic wavefield and the possibility of decomposing the displacement fields into Pand S-wave modes (Etgen, 1988; Zhe and Greenhalg, 1997; Yan and Sava, 2007; Yan and Xie, 2010; Duan and Sava, 2014a). Correlating the displacement fields for each component of the source and receiver wavefields leads to images with a mixture of Pand S-wave modes, thus making interpretation challenging. In addition, the nine images generated by this kind of imaging condition represent another difficulty for 3D interpretation. Alternatively, if the displacement wavefields are separated into Pand S-waves using Helmholtz decomposition, one can correlate specific wave modes from source and receiver wavefields (Etgen, 1988; Yan and Sava, 2007). Another issue for displacement and potential imaging conditions is polarity reversal, which occurs in elastic images due to changes in the elastic wavefield polarization. Specifically, converted waves change sign due to the different orientation of Pand S-polarization vectors in relation to subsurface interfaces. For isotropic media, this polarity reversal occurs at normal incidence (Balch and Erdermir, 1994), which enables polarity reversal corrections either after angle-domain imaging (Yan and Sava, 2008) or by exploiting the relationship between incidence directions and reflector orientation (Duan and Sava, 2014a). In addition to the issue of polarity reversal, elastic images have the disadvantage of containing artifacts that degrade image quality. Manuscript received by the Editor 14 September 2015; revised manuscript received 11 January 2016; published online 30 May 2016. Center for Wave Phenomena, Colorado School of Mines, Golden, Colorado, USA. E-mail: [email protected]; [email protected]. Z-Terra Inc., Houston, Texas, USA. E-mail: [email protected]. © 2016 Society of Exploration Geophysicists. All rights reserved. S207 GEOPHYSICS, VOL. 81, NO. 4 (JULY-AUGUST 2016); P. S207–S219, 11 FIGS. 10.1190/GEO2015-0487.1 D ow nl oa de d 07 /1 8/ 16 to 1 38 .6 7. 12 .1 84 . R ed is tr ib ut io n su bj ec t t o SE G li ce ns e or c op yr ig ht ; s ee T er m s of U se a t h ttp :// lib ra ry .s eg .o rg / As shown in detail elsewhere (Yan and Sava, 2007; Ravasi and Curtis, 2013; Duan and Sava, 2014a), injecting elastic data into a model (implementing back propagation) creates the so-called fake modes during the wavefield extrapolation step and leads to artifacts during the application of an imaging condition. These artifacts might be present even after stacking, masking weak reflections in the final image (Duan and Sava, 2014b). A second type of artifact appears if the elastic model contains sharp interfaces. In this case, backscattered reflections occur during the wavefield extrapolation step, and the application of conventional imaging conditions create lowwavenumber artifacts in the image (Youn and Zhou, 2001; Yoon and Marfurt, 2006; Guitton et al., 2007; Denli and Huang, 2008; Chen and Huang, 2014). Here, we address the second type of artifact; the first type is beyond the scope of this paper because it requires nonconventional data acquisition or additional assumptions. Considering the issue of polarity reversal and backscattering artifacts, and the fact that several images for different wave modes are difficult to interpret, we seek an imaging condition that outputs an attribute of the earth’s reflectivity into a single image. This single image should facilitate interpretation and provide a concise description of the imaged structures. In addition, we intend to assign a clear physical explanation to images obtained with this imaging condition. We propose a new imaging condition that captures all wave modes into a single image, with attenuated backscattering artifacts and without polarity reversal at normal incidence. Our imaging condition is derived from the energy conservation principle of an elastic wavefield. THEORY We use energy conservation laws analogous to the acoustic case (Rocha et al., 2015) to derive a function that measures the energy of a wavefield. This energy function also enables us to form an imaging condition for two extrapolated elastic wavefields from source and receivers. Wavefield extrapolation For an isotropic medium enclosed by a domain Ω ⊂ R, we can write a wave equation for the displacement vector with no external sources (Aki and Richards, 2002): ρÜ 1⁄4 ∇1⁄2λð∇ · UÞ þ ∇1⁄2μ · ð∇Uþ ∇UTÞ : (1) In equation 1, the elastic wavefield is a function of space x, time t, and experiment index e: Uðe; x; tÞ for x ∈ Ω and t ∈ 1⁄20; T . The dot symbol indicates dot product, the superscript dot indicates the first time differentiation, and the double dot indicates the second time differentiation. Because elastic waves propagate at a finite speed, we can take the spatial domain Ω large enough, and then assume homogeneous boundary conditions for U and its derivatives, which means that U and its derivatives vanish on the boundary of Ω (∂Ω) for all t ∈ 1⁄20; T . The density and the Lamé parameters are functions of space ρðxÞ, λðxÞ, and μðxÞ, and we assume these parameters vary slowly inΩ, which allows us to neglect their spatial derivatives, thus leading to a different wave equation: ρÜ 1⁄4 λ1⁄2∇ð∇ · UÞ þ μ1⁄2∇ · ð∇Uþ ∇UTÞ ; (2) and equation 2 can be used to extrapolate elastic wavefield for arbitrary sources, assuming that we know the spatial distribution of parameters ρ, λ, and μ. Elastic wavefield energy The total energy of an acoustic wavefield is conserved; i.e., we can write (Appendix A)