On the true-amplitude imaging condition for reverse-time migration

The true-amplitude (TA) imaging condition for reverse-time migration (RTM) is based on a combination of temporal and spatial derivatives of the upand downgoing wavefields. By means of partial integrations (or redistribution of the frequency factors in the frequency domain, we derive several alternative expressions for this imaging condition. Interestingly, the temporal derivatives can be completely replaced by spatial derivatives and temporal integrations. In this way, one version of the TA imaging condition makes use of the Laplacian operator, in this way relating to a common way of removing backscattering artifacts in RTM. We demonstrate by means of numerical examples using the Marmousi and Sigsbee2A data that the quality of the migrated image strongly depends on the version chosen for implementation. The best quality is achieved with a version that combines second derivatives of the source wavefield with the Laplacian operator. INTRODUCTION Reverse-time migration (RTM) is a seismic imaging method based on the full (two-way) wave equation (Schultz and Sherwood, 1980; McMechan, 1983; Baysal et al., 1983; Sun and McMechan, 2001; Yan and Sava, 2008). In the same way as other wave-equation based migration techniques, it makes use of an image condition, the most basic form of which is simple cross-correlation of the upand downgoing wavefields (Claerbout, 1971). In the early days of seismic imaging, RTM was not of much practical use because of its high computational cost and the presence of strong low-frequency artifacts from backscattering if the velocity model contains sharp velocity contrasts. Its use has gained much popularity in the first decade of this century, after the advent of more powerful computers and new technologies to remove the backscattering artifacts (Yoon et al., 2004; Fletcher et al., 2006; Guitton et al., 2006). Most of these techniques rely on modified imaging conditions (see, e.g., Yoon and Marfurt, 2006; Costa et al., 2009; Luo et al., 2009, 2010; Zhu et al., 2009). In the same context, Kiyashchenko et al. (2007) and Op’t Root et al. (2012) derive the true-amplitude (TA) imaging condition for reverse-time migration (RTM). The purpose of the TA imaging condition is to remove the backscattering artifacts and to provide image amplitudes that are proportional to reflection coefficients. Unfortunately, in the form presented by these authors, the TA imaging condition does not allow for an efficient implementation in the time domain. For a time-domain implementation, it must be recast into a different form. In this paper, we derive a number of theoretically equivalent forms of the approximate TA imaging condition that can be efficiently implemented in the time domain. In a similar way to Douma et al. (2010), we show that the true-amplitude imaging condition for RTM can be reformulated into a version containing the Laplacian operator. This operator is frequently used in seismic imaging without a profound theoretical basis to remove the low-frequency backscattering artifacts from RTM images. We numerically evaluate the derived time-domain versions of the TA imaging condition by comparing the resulting migrated images of the Marmousi and Sigsbee2A data. 16 Annual WIT report 2019 TRUE-AMPLITUDE IMAGING CONDITION According to Op’t Root et al. (2012), the TA imaging condition is given in the frequency domain by Ir(x) = 1 2π ∑ s ∫ ω dω 1 (−iω)PsPs [ PsPr − c(x) ω2 ∇Ps · ∇Pr ] , (1) where Ps = Ps(ω,x;xs) and Pr = Pr(ω,x;xs) are the (downgoing) source and (upgoing) receiver wavefields for a source at xs, downward continued to the imaging point x. Moreover, the bar over a symbol denotes the complex conjugate operation. Equation (1) is slightly different from the one of Op’t Root et al. (2012). For simplicity, we have assumed that the source wavelet is a (possibly band-limited) delta-function, the effects of which are acceptable in the final migrated image. Therefore, we have combined in equation (1) the Green’s function and source wavelet in the formula of Op’t Root et al. (2012) into the source wavefield Ps. Moreover, we have made the sum over all sources explicit. Finally, the different sign of the factor (−iω) in the denominator of the above equation results from our use of the following definition of the Fourier transform pair, f(ω) = ∫ ∞ −∞ dt f(t)e e f(t) = 1 2π ∫ ∞ −∞ dω f(ω)e−iωt . (2) When trying to implement the TA imaging condition in the time domain, one recognizes that its basic form, equation (1) is not very favorable. For an efficient time-domain implementation, it must be recast into a more adequate form. Below, we derive a number of theoretically equivalent forms. We then compare them numerically by looking at the corresponding images. Implementational forms The advantage of the frequency domain representation of the TA imaging condition in equation (1) is that the time derivatives are represented by factors (−iω). Thus, it immediately allows us to recognize that these factors can be rather freely redistributed among the wavefield terms. Making use of this freedom, our first rewrite moves the ω in the denominator of the spatial-derivatives term, where it would represent a two-fold time integration, to a position in front of the parentheses. This results in Ir(x) = ∑ s ∫ ω dω 1 (−iω)(−ω2)PsPs [ −ωPsPr + c(x)∇Ps · ∇Pr ] ,