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Compensating for visco‐acoustic effects in reverse‐time migration
... Dai and West [1], Mittet et al. [2], and Wang [3] utilized the one-way wave-equation migration method to compensate for both the frequency-dependent absorption and phase distortion by introducing the frequency-dependent complex wavenumber associated with the attenuation model. In the context of reverse time migration (RTM), Zhang et al. [4] derived a viscoacoustic wave equation in the time domain using the pseudo-differential operator and applied it to the Q-compensated RTM (Q-RTM) method for compensating the attenuation effects. Suh et al. [5] extended the viscoacoustic wave equation developed by Zhang et al. [4] to the anisotropic case and validated the effectiveness of Q-RTM in the vertical transversely isotropic media. ...
... In the context of reverse time migration (RTM), Zhang et al. [4] derived a viscoacoustic wave equation in the time domain using the pseudo-differential operator and applied it to the Q-compensated RTM (Q-RTM) method for compensating the attenuation effects. Suh et al. [5] extended the viscoacoustic wave equation developed by Zhang et al. [4] to the anisotropic case and validated the effectiveness of Q-RTM in the vertical transversely isotropic media. Zhu et al. [6], [7] derived a new viscoacoustic wave equation in the time domain using fractional spatial derivatives and applied it to the Q-RTM method. ...
... In the Q-compensated wave-equation migration methods, they generally need a stabilization step to stabilize the time-reversal receiver wavefield, because the amplification of the high-frequency wavefield which increases with the extrapolation time-step or depth-step will lead to the numerical instability problem [8], [9], [10], [11], [12], [13]. The low-pass filter [6], [7], the regularization scheme [4], [5], the superseded attenuation coefficients [10], [11], and the stable compensation operator using a stable division [12], [13] schemes can be used to mitigate this numerical instability problem. ...
Image-domain least-squares reverse time migration (IDLSRTM) through point spread functions (PSFs) is a suitable compromise between image quality and computational efficiency for inversion-based imaging tools. However, the conventional IDLSRTM method in acoustic approximation does not account for the subsurface attenuation effects, which may result in the unfocused migration image in attenuated geological environments. To incorporate the attenuation effects and improve the image quality, we develop a Q-compensated IDLSRTM method by using the hybrid PSFs rather than the acoustic PSFs as the blurring functions to deconvolve the adjoint migration image. These hybrid PSFs are estimated by a combination of computation between the viscoacoustic Born modeling and acoustic reverse time migration (RTM) using a series of uniform point scatterers. To further improve the quality of inverted images, we have applied a hybrid deblurring filter to the hybrid PSFs and acoustic RTM image, before the iterative inversion. Through some numerical examples of synthetic and field data, we have demonstrated that the proposed Q-IDLSRTM method combined with the hybrid PSFs and the hybrid deblurring filter can compensate for the attenuation effects and provide seismic images with improved spatial resolution and balanced image amplitudes. Relative to the conventional IDLSRTM methods through acoustic and hybrid PSFs, the proposed method can provide migration images with higher image resolution and better-balanced image amplitudes.
... The second approach is based on wave equations and compensates for attenuation along the propagation path (Dai and West, 1994;Mittet, 2007;Li et al., 2016b). Since the attenuation occurs during wave propagation, it is more reasonable to implement Q-compensated as part of the migration process (Zhang et al., 2010;Zhu and Sun, 2017;Zhao Y. et al., 2018). ...
... Several strategies have been proposed to improve numerical stability. For example, the schemes include regularization approaches (Zhang et al., 2010;Wang et al., 2012), filter-based approaches Wang Y. et al., 2018;Chen et al., 2020a), improved imaging conditions (Xie et al., 2015;Zhao X. H. et al., 2018;Sun and Zhu, 2018;Yang et al., 2021) and least-squares Q reverse-time migration (QLSRTM) (Chen et al., 2020b;. Among these, the implicit adaptive stabilization compensation scheme (Wang et al., 2017) that adjusts the truncation frequency according to the propagation time and Q value provides a better trade-off between numerical stability and imaging resolution. ...
Prestack reverse-time migration (RTM) is a popular imaging technique for complex geological conditions, since the amplitude attenuation and velocity dispersion are common in seismic recordings. To image attenuated seismic recordings accurately, a robust migration algorithm with a stable attenuation compensation approach should be considered. In the context of the Q-compensated RTM approach based on the decoupled fractional Laplacians (DFLs) viscoacoustic wave equation, amplitude compensation can be implemented by flipping the sign of the dissipation term. However, the non-physical magnification of image amplitude could lead to a well-known numerical instability problem. The explicit stabilization operator can rectify the amplitude attenuation and suppress the numerical instability. However, limited by the inconvenient mixed-domain operator, the average Q value rather than the real Q value is often used in the compensation operator, lowering the compensated accuracy of the migration image. To overcome this problem, we propose a novel explicit Q-compensation scheme. The main advantage of the proposed compensation operator is that its order is space-invariant, making it more suitable for handling complex heterogeneous attenuation media. Several two-dimensional (2D) and three-dimensional (3D) synthetic models are used to verify the superiority of the proposed approach in terms of amplitude fidelity and image resolution. Field data further demonstrates that our approach has potential applications and can greatly enhance the resolution of seismic images.
... Thus, these approaches enjoy greater simplicity and higher efficiency at the expense of the compensation accuracy. However, the attenuation effects occur during seismic wave propagation, so it is more physically reasonable to carry out attenuation compensation in the migration [22]. Therefore, the propagation-path-based compensation methods implemented during the seismic wave propagation have also been investigated by many researchers [23][24][25]. ...
... Among them, each one has its own advantages and disadvantages. For example, the OWEM is commonly formulated in the frequency domain; therefore, Q-OWEM can be easily implemented by replacing the real-value velocity with the complex-value phase velocity [22]. Although Q-OWEM is easy to implement, it has some limitations to image the complex geological structures [33]. ...
Seismic attenuation occurs during seismic wave propagation in a viscous medium, which will result in a poor image of subsurface structures. The attenuation compensation by directly amplifying the extrapolated wavefields may suffer from numerical instability because of the exponential compensation for seismic wavefields. To alleviate this issue, we have developed a stabilized frequency-domain Q-compensated reverse time migration (FQ-RTM). In the algorithm, we use a stabilized attenuation compensation operator, which includes both the stabilized amplitude compensation operator and the dispersion correction operator, for the seismic wavefield extrapolation. The dispersion correction operator is calculated based on the frequency-domain dispersion-only viscoacoustic wave equation, while the amplitude compensation operator is derived via a stabilized division of two propagation wavefields (the dispersion-only wavefield and the viscoacoustic wavefield). Benefiting from the stabilization scheme in the amplitude compensation, the amplification of the seismic noises is suppressed. The frequency-domain cross-correlation imaging condition is exploited to obtain the compensated image. The layered model experiments demonstrate the effectiveness and great compensation performance of our method. The BP gas model examples further verify its feasibility and stability. The field data applications indicate the practicability of the proposed method. The comparison between the acoustic and compensated results confirms that the proposed method is able to compensate for the seismic attenuation while suppressing the amplification of the high-frequency seismic noise.
... To compensate for the energy attenuation in seismic data, inverse Q filtering is developed (Bickel and Natarajan, 1985;Hargreaves and Calvert, 1991;Li and Wang, 2007). However, the theoretical basis of inverse Q filtering is wave propagation in a one-dimensional subsurface medium model, which is not applicable to inhomogeneous complex geological conditions (Zhang et al., 2010). Based on the continuous development of ray theory, one-way wave equation, and two-way wave equation methods, more accurate inverse Q migration algorithms are gradually developed in the field of prestack depth migration. ...
... During the backward propagation of the viscoacoustic seismic wavefield, the rapid growth of high-frequency components energy can lead to instability of the numerical results. To solve this problem, scholars have proposed methods such as adding regularization terms (Zhang et al., 2010;Tian et al., 2015;Zhao et al., 2018), and low-pass filtering (Bai et al., 2013;Zhu et al., 2014). However, none of these methods can fundamentally solve the instability problem. ...
With the deepening of oil and gas exploration and the increasing complexity of exploration targets, the influence of anisotropy and anelasticity of subsurface media on seismic imaging cannot be ignored. The least-squares reverse time migration is developed on the idea of linear inversion, which can effectively solve the amplitude imbalance, low resolution, and serious imaging noise problems of RTM. In this paper, based on the viscoacoustic pure qP-wave equation, the corresponding demigration operator, adjoint operator, and gradient-sensitive kernel are derived, and the least-square reverse time migration imaging algorithm of viscoacoustic pure qP-wave in VTI medium is proposed. During iterative inversion, the inverse of Hessian is approximately solved to achieve stable attenuation compensation. Finally, we verify the effectiveness and applicability of the proposed viscoacoustic VTI least-squares reverse time migration imaging algorithm through the model tests and field data. The numerical results show that the method can compensate for the amplitude loss and phase distortion caused by attenuation, and correct the anisotropy-induced misalignment of the reflection interfaces, which improves the accuracy and resolution of the imaging profile.
... After that, Deng and Mcmechan (2008) further extend their method into visco-elastic media. Zhang et al. (2010) derive a fractional visco-acoustic WE for Q-RTM and stabilize the algorithm using a regularization process. Bai et al. (2013) implement Q-RTM using a visco-acoustic WE without memory variables, which is derived from the relaxation function of SLS model. ...
... Therefore, in recent years, more and more attenuations have been paid to studying the migration method for attenuation compensation in anisotropic media. Combining Fletcher et al. (2009) andZhang et al. (2010), Suh et al. (2012) derive a coupled second-order fractional visco-acoustic transversely isotropic (TI) WE and apply it to Q-RTM using synthetic and field data. Based on the relaxation function of SLS model, Xu et al. (2015a) and Xu et al. (2015b) formulate the visco-acoustic pure qP-wave vertical transversely isotropic (VTI) and TTI WEs respectively, and then they implement Q-RTM in TI media using the above equations. ...
The anelasticity and anisotropy widely exist in real subsurface media. Strong anelasticity will lead to phase dispersion and amplitude attenuation during seismic wave propagation. Anisotropy cause seismic waves to have obviously different kinematic characteristics from that of isotropy. For seismic migration, ignoring the anelasticity and anisotropy of subsurface media will significantly reduce the quality of migration images, even cause imaging failure. We propose a Q-compensated least-squares reverse time migration (Q-LSRTM) in tilted transversely isotropic (TTI) media to correct these effects. According to the Born approximation of seismic wave equation, a linearized visco-acoustic TTI pure qP-wave modeling operator is derived using a new visco-acoustic TTI wave equation for one standard linear solid (SLS) model, which can deal with the anelasticity and can simulate pure qP-wave steadily in attenuating anisotropic media without qSV wave artifacts. Then, the corresponding adjoint equation is formulated using the adjoint-state method to calculate the gradient sensitivity kernel for the visco-acoustic TTI media. Because of the least-squares inversion, the Q compensation can be achieved during the iterations, so that the over-amplification of noises can be avoided naturally. In addition, compared with conventional LSRTM, the proposed method compensates for the anelasticity and corrects the anisotropy, so as to produce images with better spatial resolution and amplitude fidelity. Numerical examples demonstrate the feasibility and advantages of the proposed method for the data including strong attenuation effects over conventional LSRTM.
... Many laboratory and field observations show that the anelastic properties of earth media cause seismic energy attenuation and waveform distortion, especially in an area with strong attenuation (Kolsky, 1956;McDonal et al., 1958;Futterman, 1962;Aki and Richards, 1980;Zhu and Harris, 2015). For example, the presence of gas chimneys in shallow strata can strongly attenuate the energy of seismic compression waves (Zhang et al., 2010;Suh et al., 2012). As a result, the migration profile becomes dim and the imaging resolution is reduced, if the anelastic properties are not compensated. ...
... In the early days, the inverse Q-filter was often used to compensate for seismic attenuation (Bickel and Natarajan, 1985;Hargreaves and Calvert, 1991;Wang, 2002Wang, , 2006. However, it is difficult to cope with real complex geologic situations because the inverse Q-filter method is based on 1D back wave propagation (Zhang et al., 2010). Considering that seismic energy compensation should be processed during wave propagation, inverse Q-migration methods using ray tracing (Traynin and Reilly, 2008;Fletcher et al., 2012;Dutta and Schuster, 2016) and one-way wave propagators (Dai and West, 1994;Yu et al., 2002;Zhang et al., 2013) are proposed. ...
We propose a new time-domain viscoacoustic wave equation for simulating wave propagation in anelastic media. The new wave equation is derived by inserting the complex-valued phase velocity described by the Kjartansson attenuation model into the frequency-wavenumber domain acoustic wave equation. Our wave equation includes one second-order temporal derivative and two spatial variable-order fractional Laplacian operators. The two fractional Laplacian operators describe the phase dispersion and amplitude attenuation effects, respectively. To facilitate the numerical solution for the proposed wave equation, we use the arbitrary-order Taylor series expansion (TSE) to approximate the mixed domain fractional Laplacians and achieve the decoupling of the wavenumber and the fractional order. Then the proposed viscoacoustic wave equation can be directly solved using the pseudospectral method (PSM). We adopt a hybrid pseudospectral/finite-difference method (HPSFDM) to stably simulate wave propagation in arbitrarily complex media. We validate the high accuracy of the proposed approximate dispersion term and approximate dissipation term in comparison with the accurate dispersion term and accurate dissipation term. The accuracy of numerical solutions is evaluated by comparison with the analytical solutions in homogeneous media. Theory analysis and simulation results show that our viscoacoustic wave equation has higher precision than the traditional fractional viscoacoustic wave equation in describing constant- Q attenuation. For a model with Q < 10, the calculation cost for solving the new wave equation with TSE HPSFDM is lower than that for solving the traditional fractional-order wave equation with TSE HPSFDM under the high numerical simulation precision. Furthermore, we demonstrate the accuracy of HPSFDM in heterogeneous media by several numerical examples.
... Seismic waves are subject to nonelastic effects while traveling through the subsurface (e.g., amplitude decay and velocity dispersion). This phenomenon has been extensively studied in laboratory experiments (Guo and Fu, 2007), global seismology (Dalton et al., 2009;Iritani et al., 2010), and seismic exploration (Zhang et al., 2010;Dutta and Schuster, 2014;Xiao et al., 2018). In exploration geophysics, attenuation in poorly consolidated shallow sediments cannot be ignored. ...
... Here, the reference frequency ω 0 is defined as 1∕t 0 , sgnð·Þ denotes the sign function. By substituting equation 4 into the wavenumber expression k ¼ ω ffiffiffiffiffiffiffiffiffi ffi ρ∕M p , we obtain the following dispersion relation (Zhang et al., 2010, see Appendix A for the derivation): ...
We present a concise time-domain wave equation to accurately simulate wave propagation in visco-acoustic media. The central idea behind this work is to dismiss the negative frequency components from a time-domain signal by converting the signal to its analytic format. The negative frequency components of any analytic signal are always zero, meaning we can construct the visco-acoustic wave equation to honor the relaxation property of the media for positive frequencies only. The newly proposed complex-valued wave equation (CWE) represents the wavefield with its analytic signal, whose real part is the desired physical wavefield, while the imaginary part is the Hilbert transform of the real component. Specifically, this CWE is accurate for both weak and strong attenuating media in terms of both dissipation and dispersion and the attenuation is precisely linear with respect to the frequencies. Besides, the CWE is easy and flexible to model dispersion-only, dissipation-only or dispersion-plus-dissipation seismic waves. We have verified these CWEs by comparing the results with analytical solutions, and achieved nearly perfect matching. Except for the homogeneous Q media, we have also extended the CWEs to heterogeneous media. The results of the CWEs for heterogeneous Q media are consistent with those computed from the nonstationary operator based Fourier Integral method and from the Standard Linear Solid (SLS) equations.
... Because wavefield attenuation occurs during the wave propagation process, it is natural and physically more accurate to correct for attenuation effects in prestack depth migration (Xie et al., 2009;Zhang et al., 2010;Valenciano et al., 2011;. Ray-based migrations (Xie et al., 2009;Xin and Hung, 2009;Bai et al., 2016) can be generalized to viscoacoustic media by incorporating an attenuation traveltime, which are integrated along rays traced in the spatial-variant Q model. ...
... One-way wave-equation migrations are commonly implemented in the frequency domain, in which amplitude and phase corrections can be implemented by reversing the imaginary part of the complex-valued velocity during the extrapolation of receiver wavefields (Mittet et al., 1995;Yu et al., 2002;Zhang and Wapenaar, 2002;Mittet, 2007;Zhang et al., 2013). Incorporating attenuation compensation into reverse time migration (RTM) can be implemented using different rheological models, including the standard linear solid model (Carcione, 2007;Deng and McMechan, 2008) and the constant-Q model (Zhang et al., 2010;. derive a viscoacoustic RTM using a complex-valued time-domain viscoacoustic wave equation. ...
Because of amplitude decay and phase dispersion of seismic waves, conventional migrations are insufficient to produce satisfactory images using data observed in highly attenuative geologic environments. We have developed a least-squares Gaussian beam migration method for viscoacoustic data imaging, which not only can compensate for amplitude decay and phase dispersion caused by attenuation, but also can improve image resolution and amplitude fidelity through linearized least-squares inversion. We represent viscoacoustic Green’s function by a summation of Gaussian beams, in which an attenuation traveltime is incorporated to simulate or compensate for attenuation effects. Based on the beam representation of the Green’s function, we construct the viscoacoustic Born forward modeling and adjoint migration operators, which can be effectively evaluated by a time-domain approach based on a filter-bank technique. With the constructed operators, we formulate a least-squares migration scheme to iteratively solve for the optimal image. Numerical tests on synthetic and field datasets demonstrate that the proposed method can effectively compensate for the attenuation effects, and produce images with higher resolution and more balanced amplitudes than images from acoustic least-squares Gaussian beam migration.
... To solve this problem, scholars have made great efforts and proposed different methods. Zhang et al. (2010), Tian et al. (2015), and Zhao et al. (2018) added the regularized term based on the viscoacoustic wave equation to maintain the stability of the wavefield backward propagation. However, the regularized coefficient is determined empirically. ...
... In recent years, LSRTM in viscous media has received a lot of attention from geophysicists. Zhang et al. (2010), and Zhu et al. (2014) have proposed some viscoacoustic wave equations that separate the effects of phase and amplitude. Based on these viscoacoustic equations, numerous viscoacoustic LSRTM methods have been proposed by many geophysicists (Li et al. 2014;Dutta and Schuster, 2014;Chen et al. 2017;Qu et al., 2021). ...
... In addition, based on the constant Q theory [11], many researchers derived their own fractional viscoacoustic (elastic) wave equations with the amplitude attenuation and phase dispersion term decoupled [12], [13], [14], [15]. After that, the Q-compensated reverse time migration (Q-RTM) method has been developed [16], [17], [18]. ...
Viscous effects cause strong energy decay and waveform changes of seismic waves. These distortions can be corrected using
Q-
compensated reverse time migration (
Q
-RTM) algorithms, and high-resolution migration images can be obtained. However, all
Q
-RTM methods require a relatively accurate
Q
model. The traditional wave-equation
Q
tomography can invert the
Q
model by eliminating the difference in peak frequency between the observed and synthetic early arrivals. However, this approach only can be used to invert the
Q
value only for large-scale applications or on the surface. Moreover, the reflected wave can also be applied in the extended domain, but its computational efficiency is low compared to that of the early arrivals. To overcome these problems, this work proposes a new wave-equation-based
Q
inversion methodology to evaluate more accurate underground
Q
values in local domain. The proposed approach is applicable both to the early arrivals and reflected waves. Accordingly, we first transform the seismic data into the local domain using a sliding Gaussian window to alleviate the crosstalk noise in nearby seismic waves. Then, we use an improved cross correlation algorithm between the amplitude spectra of the observed and synthetic data to calculate the peak frequency shift of each seismic event in local domain. Thus, the inversion accuracy of
Q
can be improved by using different kinds of waves. The numerical inversion examples demonstrate the ability of our proposed method to produce satisfactory inversion results, especially in high-attenuation and deep areas. The
Q
-RTM images further illustrate the accuracy of our proposed
Q
tomography method.
... Nonetheless, when solving the fractional time derivative, the Grünwald-Letnikov method necessitates a large computer memory to store all previous wavefields. To avoid the large memory requirements, Zhang et al. (2010) proposed a new viscoacoustic wave equation that replaces the fractional time derivative with the spatial fractional Laplacian. proposed a CQ viscoacoustic wave equation based on fractional Laplacians that describes decoupled amplitude attenuation and phase dispersion effects. ...
The propagation of seismic waves in attenuating and anisotropic earth media is accompanied by amplitude attenuation and phase distortion. If these adverse effects are not addressed in seismic imaging, we may end up with inaccurate reflector positions, dimming amplitudes, and reduced spatial resolution in the imaging results. We use a pure pseudo-viscoacoustic TTI wave equation as a forward engine to implement Q-compensated TTI reverse time migration (RTM) because the wavefields simulated by the conventional coupled pseudo-viscoacoustic tilted transversely isotropic (TTI) wave equation contain shear wave artifacts and are unstable when the anisotropic parameters ε < δ. The high-frequency noise in the wavefield will be amplified exponentially during amplitude-compensated extrapolation, resulting in numerical instability when using Q-compensated TTI RTM. To eliminate the destabilizing effect of boosted high-frequency noise, we introduce a complex velocity that can be used to describe amplitude compensation over the limited frequency band. Then, based on this complex velocity, we derive a stable amplitude-compensated operator and apply it to the Q-compensated TTI RTM. The numerical simulation results show that, in comparison with the coupled pseudo-viscoacoustic TTI wave equation, the pure pseudo-viscoacoustic TTI wave equation is free from shear wave artifacts and is not restricted by anisotropic parameters. In addition, the pure pseudo-viscoacoustic TTI wave equation has high accuracy in describing velocity anisotropy and attenuation isotropy. Synthetic and field data examples demonstrate the effectiveness of our Q-compensated TTI RTM in compensating amplitude dissipation and correcting phase distortion.
... This approach introduces a stable compensation operator in TTI media, although the square of the division is likely to damage more medium-and high-frequency effective signals. Based on the dispersion relation in a linear viscoacoustic medium (Kjartansson, 1979), Zhang et al. (2010) present a viscoacoustic wave equation using the pseudodifferential operator in the time domain for isotropic media and apply it in RTM. Along similar lines, Bai et al. (2013) derive a new viscoacoustic wave equation in which memory variables are avoided and use this as the basis for carrying out attenuation compensation within RTM. ...
... However, it cannot completely eliminate the attenuation effect, and the stability of the inverse Q filtering has always been an important research topic (Braga and Fernando, 2013). Deng and McMechan (2007) propose reverse time migration with the Q-compensated method, which is further developed by Zhang et al. (2010). Oliveira et al. (2009) apply the analytical solution of the viscoacoustic equation to execute nonlinear wave impedance inversion and obtain high-resolution impedance parameters. ...
Amplitude variation with offset (AVO) inversion is based on single interface reflectivity equations. It involves some restrictions, such as small-angle approximation, including only primary reflections, and ignoring attenuation. To address the above mentioned shortcomings, the analytical solution of one-dimensional viscoelastic wave equation is utilized as the forward modeling engine for prestack inversion. This method can conveniently handle the attenuation and generate the full wavefield response of a layered medium. To avoid numerical difficulties of the analytical solution, the compound matrix method (CMM) is applied to rapidly obtain the analytical solution by loop vectorization. Unlike full waveform inversion (FWI), the proposed prestack waveform inversion (PWI) can be performed a target-oriented way and can be applied in reservoir study. Assuming that a Q value is known, PWI is applied to synthetic data to estimate elastic parameters including (P-and S-wave velocity and density). After validating the proposed method on synthetic data, this method is applied to a reservoir characterization case study. The results indicate that the reflectivity calculated by the proposed approach is more realistic than that computed by using single interface reflectivity equations. Attenuation is an integral effect on seismic reflection; therefore, the sensitivity of seismic reflection to P-and S-wave velocity and density is significantly greater than that to Q, and the seismic records are sensitive to the low-frequency trend of Q. Thus, we can invert for the three elastic parameters by applying the fixed low-frequency trend of Q. In terms of resolution and accuracy of synthetic and real inversion results, the proposed approach performs superior to AVO inversion.
... A successful Q-compensated wave-propagation process boosts the amplitude accurately, while maintaining the same dispersion relation as the Q-attenuation process. Several Q-RTM algorithms were proposed to simulate this wave behavior by decoupling the attenuation effects on the amplitude and the phase by modifying the governing equations of wave propagation (Zhang et al., 2010;Xie et al., 2015). These algorithms compensate the Q effects properly and improve the migration image qualities, but they are all based on Kjartansson's constant Q (KCQ) model and pseudospectral methods. ...
Q-compensated Reverse-Time Migration (Q-RTM) is effective for improving the seismic imaging quality degraded by low-Q anomalies. However, it is hard to apply existing pseudo-spectral-based Q-RTM methods to large-scale problems due to the obstacles to high-efficient parallelization posed by the global pseudo-spectral operators. On the other hand, finite-difference Q-RTM is intrinsically appropriate for domain decomposition and parallel computation, thus suitable for industrial-sized problems but facing a two-fold challenge: (1) to effectively compensate the phase in a broad-bandwidth sense during the wave back-propagation process; and 2) to accurately handle the tilted transverse isotropic (TTI) medium with attenuation. We develop a new framework of finite-difference Q-RTM algorithm by expanding the linear viscoacoustic constitutive relation to a series of integer-order differential terms and a unique integral term which can decouple the amplitude and the phase to allow accurate compensation in a broader frequency range. This framework has two typical implementations: (1) optimizing the frequency-dependent phase velocity (while fixing the negative constant Q); and (2) optimizing the Q value (while fixing the frequency-dependent phase velocity). We generalize this broadband finite-difference Q-RTM algorithm to TTI media, where an artificial Q s is applied to suppress the S-wave artifacts induced by the acoustic TTI approximation. Numerical examples demonstrate that this Q-RTM method accurately compensates both the phase and amplitude in a broad frequency range of 5-70 Hz and produces high-quality images. Due to the local nature of finite-difference operators, this algorithm is expected to outperform the existing pseudo-spectral-based Q-RTM methods in terms of computational efficiency and implementation convenience for real world Q-RTM projects.
... Traynin et al. (2008) developed Kirchhoff Q migration by introducing amplitude and phase compensating terms. Zhang et al. (2010), Zhu et al. (2014) proposed a reverse time Q reverse time migration approach using a new formulated time-domain visco-acoustic wave equation with separated operators dominating amplitude loss and phase dispersion, respectively. ...
Under the assumption of invariant ray path in a weakly dissipative (high quality factor Q) subsurface medium, a tomographic inversion approach composed of two cascading applications of first arrival traveltime and Q tomography is proposed for compensating amplitude loss caused by near-surface anomalies, such as unconsolidated soils or the overburden gas cloud. To improve the computational efficiency, these two related tomography methods were adopted with an adjoint-state technique. First, arrival traveltime tomography will be performed to provide an inverted velocity model as one of the inputs for the following first arrival Q tomography. Then, the synthetic first break generated by the inverted velocity model will be used as a stable guidance of accessing the scopes of first arrival waveforms in the time domain where the potential attenuated time information is contained. The attenuated time will be estimated through a logarithmic spectral ratio linear regression corresponding to frequency-dependent propagation responses of different wave types. All these estimated attenuated times will be applied with reference signals to generate synthetic attenuated seismic data in the time domain, and their discrepancies with real data will be evaluated using similarity coefficients. The ones with larger values will be selected as optimal attenuated time inputs for the following Q tomographic inversion. Examples of both synthetic and field data reveal the feasibility and potential of this method.
... To recover the amplitude loss while preserving the phase, a time-reversal idea (Treeby et al., 2010;Zhu, 2014) is introduced to Q-RTM to separate the operator that controls amplitude loss from the wave propagator. Zhang et al. (2010) first report a decoupled fractional Laplacian (DFL) viscoacoustic wave equation. Later, use a more rigorous method to derive a novel DFL viscoacoustic wave equation. ...
Reverse-time migration with Q compensation (Q-RTM) is an effective approach to enhance the resolution of seismic image, because it retrieves both the amplitude loss and phase distortion induced by the viscosity of media. According to the cross-correlation imaging condition, Q-RTM requires compensation for the amplitude loss in the propagation paths of both source and receiver wavefields, which can be realized by solving an amplitude-boosted wave equation. However, the amplitude-boosted simulations suffer from numerical instability due to the amplification of high-frequency noise. We have developed a robust stabilization strategy for Q-RTM by incorporating a time-variant filter into the amplitude-boosted wavefield extrapolation step. We modify the Fourier spectrum of the operator that controls the amplitude compensation to be time variant, and add to the spectrum a stabilization factor. Doing so, we integrate the time-variant filter into the viscoacoustic wave-propagator implicitly, and avoid any explicit filtering operation in Q-RTM. We verify the robustness of this stabilized Q-RTM with two synthetic data examples. We also apply this technique to a field dataset to demonstrate the imaging improvements compared to an acoustic RTM and a more traditional Q-RTM method.
Seismic waves propagating through attenuating media induce amplitude loss and phase dispersion. Neglecting the attenuation effects during seismic processing results in the imaging profiles with weakened energy, mispositioned interfaces and reduced resolution. To obtain high‐quality imaging results, Q ‐compensated reverse time migration is developed. The kernel of the Q ‐compensated reverse time migration algorithm is a viscoacoustic wave equation with decoupled amplitude loss and phase dispersion terms. However, the majority of current decoupled viscoacoustic wave equations are solved using the computationally expensive pseudo‐spectral method. To enhance computational efficiency, we initiate our approach from the dispersion relation of a single standard linear solid model. Subsequently, we derive a novel decoupled viscoacoustic wave equation by separating the amplitude loss and phase dispersion terms, previously coupled in the memory variable. The newly derived decoupled viscoacoustic wave equation can be efficiently solved using the finite‐difference method. Then, we reverse the sign of the amplitude loss term of the newly derived viscoacoustic wave equation to implement high‐efficient Q ‐compensated reverse time migration based on the finite‐difference method. In addition, we design a regularization term to suppress the high‐frequency noise for stabilizing the wavefield extrapolation. Forward modelling tests validate the decoupled amplitude loss and phase dispersion characteristics of the newly derived viscoacoustic wave equation. Numerical examples in both two‐dimensional and three‐dimensional confirm the effectiveness of the Q ‐compensated reverse time migration based on the finite‐difference algorithm in mitigating the attenuation effects in subsurface media and providing high‐quality imaging results.
The strong viscosity of the subsurface introduces amplitude absorption and phase-velocity dispersion. Incorrect compensation of the inherent attenuation (the strength of seismic attenuation can be quantified by the inverse of quality factor Q, which is defined as 2π times the ratio of the stored energy to the lost energy in a single cycle of deformation) can significantly affect imaging quality. While Q-least squares reverse time migration allows for the compensation of attenuation effects during the iterations, the traditional L 2 -norm-minimization, which is highly sensitive to the source wavelet, poses a challenge in accurately estimating source wavelet from field data. Thus, we develop a source-independent Q-least squares reverse time migration, in which a convolutional objective function is introduced to replace the L 2 -norm constraint in order to mitigate the source wavelet effect. According to the Born approximation, we first linearize the constant-order decoupled fractional Laplacian viscoacoustic wave equation to derive the demigration operator, then construct the corresponding adjoint equation and gradient based on the convolutional objective function, iteratively estimating the reflectivity images. The proposed method relaxes the sensitivity to the wavelet compared to the conventional L 2 -norm scheme due to the convolutional objective function, which has the ability to construct the same new source for simulated and observed data. Numerical tests on a layered model, the Marmousi model, and field data demonstrate that the proposed source-independent Q-least squares reverse time migration enables us to obtain high quality reflectivity images even when using incorrect source wavelets.
The pseudo-differential operator with symbol |k|α has been widely used in seismic modeling and imaging when involving attenuation, anisotropy and one-way wave equation, which is usually calculated using the pseudo-spectral method. For large-scale problems, applying high-dimensional Fourier transforms to solve the wave equation that includes pseudo-differential operators is much more expensive than finite-difference approaches, and it is not suitable for parallel computing with domain decomposition. To mitigate this difficulty, we present a truncated space-domain convolution method to efficiently compute the pseudo-differential operator , and then apply it to viscoacoustic reverse-time migration. Although is theoretically nonlocal in the space domain, we take the limited frequency band of seismic data into account, and constrain the approximated convolution stencil to a finite length. The convolution coefficients are computed by solving a least-squares inverse problem in the wavenumber domain. In addition, we exploit the symmetry of the resulting convolution stencil and develop a fast spatial convolution algorithm. The applications of the proposed method in Q-compensated reverse-time migration demonstrate that it is a good alternative to the pseudo-spectral method for computing the pseudo-differential operator , with almost the same accuracy but much higher efficiency.
Full waveform inversion (FWI) has been proven as an effective method to estimate subsurface parameters by iteratively reducing the data residual between the predictions and the observations. Nevertheless, FWI is greatly dependent on the initial model and a poor initial model will lead to a wrong solution. Furthermore, owing to the anelasticity of the earth, seismic waves will attenuate during propagation, which results in an attenuated gradient and makes the convergence rate of FWI even worse in viscoacoustic media. To mitigate these problems, we propose an improved method for multiparameter (e.g. velocity and Q) waveform inversion. Benefiting from the theory of Q-compensated wavefield propagation, we formulate a Q-compensated joint multiparameter waveform inversion method to weaken the nonlinearity of the FWI objective function, which enables it to cope with challenges related with attenuation-induced gradient energy loss and cycle skipping simultaneously. We refer to the proposed Q-compensated joint multiparameter FWI scheme as QJMFWI. The main contributions of QJMFWI are: (1) given the difficulty associated with the estimating of velocity and Q simultaneously in viscoacoustic media, QJMFWI provides a straightforward waveform inversion method for velocity and Q model construction, by which we can obtain velocity and Q information with improved accuracy and resolution; (2) compared with conventional FWI methods, QJMFWI relaxes the requirement for good initial velocity and Q model, which can avoid trapping into local minima. Numerical and field data examples demonstrate that QJMFWI is an effective method to invert for accurate subsurface parameters in viscoacoustic media.
Least-squares reverse-time migration (LSRTM) has become an advanced technique for complex structures imaging of the subsurface, as it can provide a higher resolution and more balanced amplitude migrated image than conventional reverse-time migration (RTM). However, the intrinsic attenuation of subsurface introduces amplitude attenuation and phase dispersion of seismic wavefield, which leads to the inverted image kinematically and dynamically inexactitude. Moreover, the imperfect geometry, limited bandwidth of seismic data, and inappropriate modeling kernel etc., would inevitably introduce two side-effects in migrated image, resulting in degradation of LSRTM imaging potential. To alleviate above issues, we present a data-domain sparsity constraint viscoacoustic least-squares reverse-time migration algorithm in this paper. In particular, we utilize the decoupled constant Q fractional Laplacians (DFLs) viscoacoustic wave equation as the modeling kernel to describe the attenuation effects of the subsurface, while a model constraint constructed in the misfit function via L1-2 norm is carried out to clear the migrated artefacts and boost the imaging resolution. Thanks to the excellent performance in sparsity, the drawbacks of unconstraint LSRTM Downloaded from https can be effectively mitigated by the L1-2 norm-based regularization. In this paper, we adopt the alternating direction of multipliers method (ADMM) to iteratively address the constrained L1-2 minimization problem by implementing a proximal operator, and three synthetic examples are hired to evaluate the effectiveness and practicability of the proposed strategy. Migration results prove that the proposed scheme can effectively compensate the attenuation effects, improve the resolution, and suppress the migration artifacts of inverted images even in the complex imaging situations.
In viscoacoustic least‐squares reverse time migration methods, the reflectivity image associated with the Q factor is negligible, inverting only the velocity (v) parameter or v related variables such as squared slowness or bulk modulus. However, the Q factor influences the amplitude and phase of the seismic data, especially in basins containing gas reservoirs or storing CO2. Therefore, the Q factor and its associated parameters must be considered in the context of viscoacoustic least‐squares reverse time migration. Thus, we propose a multiparameter viscoacoustic least‐squares reverse time migration procedure, which obtains the inverse of bulk modulus () and the Q magnitude () simultaneously. We derive and implement the multiparameter forward and adjoint pair Born operators and the gradient formulas concerning and parameters. Then, we apply these derivations in our proposed multiparameter approach, which can produce images with better balanced amplitudes and more resolution than conventional reverse time migration images.
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The conventional time migration method does not consider the attenuation caused by the viscoelasticity of the underground media during the imaging process. Therefore, the final imaging amplitude and phase include inaccuracies caused by attenuation. In this study, we developed a migration scheme to compensate for absorption and dispersion using an effective quality factor (Q) estimation based on a back-propagation (BP) neural network. We used BP neural network technology to automatically estimate the effective Q value from stacked imaging data, thereby avoiding manual Q estimation using conventional methods. The proposed scheme can be incorporated into conventional seismic data-processing workflows. Furthermore, synthetic and field datasets were used to validate the proposed scheme, which was used to acquire high-resolution images with low noise levels. In addition to developing a completely data-driven Q-value estimation strategy, this study demonstrated close integration of artificial intelligence, data mining, and conventional geophysics; the proposed approach is appropriate for estimating the effective Q and has strong industrial application value and significance.
Intrinsic attenuation in seismic wave propagation leads to amplitude dissipation and phase dispersion in seismic data. The attenuation effect is a common cause for inaccurate image locations and dimmed image energy in reverse time migration (RTM). Conventional viscoacoustic or viscoelastic RTM (QRTM) methods implement attenuation compensation by reversing the sign of the amplitude loss term and keeping the dispersion term unchanged for both forward and backward wavefield extrapolation, which requires decoupled viscoacoustic wave equation and gives rise to numerical instability issue. To address these problems, we have developed a stable and effective two-way attenuation compensated viscoacoustic RTM method. The proposed method uses a new two-way normalized crosscorrelation imaging condition to compensate the attenuation effect. In the new imaging condition, only attenuated forward wavefield of sources and the viscoacoustic Greens functions of receivers are required, which eliminates the instability source arising from the operation of reversing the sign of dissipation term in viscoacoustic wave equation. Since no modifications are made to viscoacoustic wave equation, the new imaging condition can be calculated without any additional computational cost and can be applied to both decoupled and coupled viscoacoustic wave equation-based RTMs, which is more flexible than other attenuation compensation strategies. Two synthetic tests and one field data example are presented to demonstrate the stability and effectiveness of the proposed method.
Steeply dipping structural imaging is a great challenge due to its poor illumination. Conventional migration methods are unable to produce an accurate image of complex steeply dipping structures. The prismatic wave can improve the illumination of steeply dipping structures and is often used to improve the imaging results of such structures. Traditional elastic wave theory assumes that seismic waves do not attenuate when propagating through subsurface media. However, during seismic wave propagation, the wave energy decays exponentially due to the absorption and attenuation of the ground layer. Subsurface attenuation leads to amplitude loss and phase distortion of seismic waves, resulting in blurring of migration amplitudes when this attenuation is not taken into account during imaging. To address this issue, a frequency‐domain Q ‐compensated prismatic reverse time migration method is proposed, which derives Q ‐compensated prismatic wavefield propagation operators. In the proposed frequency‐domain Q ‐compensated prismatic reverse time migration, Q attenuation is fully compensated along three propagation paths and two propagation types of prismatic waves. The optimized four‐order mixed 25‐point difference format and LU decomposition method are used to solve the Q ‐compensated prismatic wavefield propagation equations with high computational efficiency. Numerical and field data examples demonstrate that the proposed frequency‐domain Q ‐compensated prismatic reverse time migration method can compensate for deep attenuation energy and improve the imaging resolution of steeply dipping structures.
The viscosity in the subsurface is ubiquitous. Prestack reverse time migration (RTM) based on the visco-acoustic wave equation is an accurate imaging method to compensate the attenuation. The resolution of an imaging result highly depends on the accuracy of the wave equation solution. We extend the nearly-analytic center difference (NACD) method to a visco acoustic wave equation to compensate seismic attenuation. The NACD method can achieve a fourth order accuracy in both the time and space domain. We employ both the pressure and its gradients to approximate the spatial derivatives based on stereo-modeling. Numerical simulations and dispersion analysis demonstrate that the proposed NACD has less numerical dispersion and is more accurate than the Lax-Wendroff correction (LWC) method. Thus, the NACD is less expensive in terms of CPU time and storage space. Numerical simulations show that the forward modeling of the visco-acoustic wavefield via the NACD is able to obtain more accurate wavefields compared to the conventional LWC method, and can accurately simulate the effect of the viscosity on waves in an attenuative media. When applied to RTM, the viscous effects of the subsurface can be compensated. Synthetic examples demonstrate that the attenuation compensated RTM using NACD can obtain an image with higher resolution in an attenuative media compared to that using the acoustic RTM. Furthermore, the image determined by the NACD is clearer than that of the LWC method in RTM.
Seismic wave suffers from amplitude attenuation and phase distortion when propagating in the attenuating media, thus reverse time migration (RTM) for viscous media should take the attenuation effects into consideration. Compensating for the attenuation effects in RTM may occur numerical instability because of the exponential amplification of the extrapolated wavefields. To obtain stable imaging results, we have developed a stabilized
Q
-compensated RTM (Q-RTM) in the frequency domain. This algorithm is implemented by the following steps: first, we use the Kolsky–Futterman model to derive a frequency-domain viscoacoustic wave equation, which can simulate the amplitude loss and phase dispersion effects separately. Then, we calculate the source wavefields in the viscoacoustic media. Next, treating the recorded (viscoacoustic) data as the receiver sources, we can obtain the phase-dispersion-only and viscoacoustic receiver wavefields, which can be used to construct the stabilized
Q
-compensated receiver wavefields. Finally, we apply the deconvolution imaging condition for obtaining a
Q
-compensated image. A simple anticline model and gas chimney model are used to verify the effectiveness of the proposed approach. The
Q
-compensated images for the noise-free data indicate the algorithmic stability and compensation accuracy of the proposed scheme. The noisy data tests for the gas chimney model demonstrate the good antinoise property of our method. The field data applications further prove their feasibility and practicability.
Anisotropy and Q attenuation bring great challenges to seismic wave migration. On migrated images, anisotropy creates structural and positioning errors, and Q attenuation leads to weak amplitudes and misplacement of reflectors. A 2D Q-compensated least-squares reverse time migration with velocity-anisotropy correction ( QLSRTM-VA) is proposed through the construction of velocity-anisotropic Q-compensated forward modeling, Q-compensated adjoint, and Q-attenuated demigration operators to simultaneously correct velocity-anisotropy and Q-attenuation in the migration process. The preceding operators are derived using first-order velocity-anisotropic viscoacoustic quasi-differential wave equations with variable densities, which are stable, capable of conveniently dealing with variable density media and are easy to transform between velocity-anisotropic Q-compensation and Q-attenuation versions. As exemplified by two synthetic and field data sets, our QLSRTM-VA method increases the imaging resolution, signal-to-noise ratio, and amplitude preservation in deep regions. Our method is capable of producing better images than viscoacoustic isotropic least-squares reverse time migration (LSRTM) and acoustic anisotropic LSRTM.
The increasing complexity of seismic exploration projects and the request for higher imaging resolution have driven the geophysics community to look for a sound understanding of the subsurface formation to optimize seismic structure interpretation and reservoir characterization. Crosswell seismic survey aims at obtaining higher resolution images of the interwell regions and more accurately characterizing the reservoir dynamics. However, the presence of the intrinsic seismic attenuation of rocks as seismic waves propagate through the subsurface results in amplitude decay and velocity dispersion. This inevitably decreases the imaging resolution and the reliability of the subsequent seismic interpretation and reservoir characterization. To compensate for the attenuating effect, one may restore to attenuation compensation technique during seismic imaging. We here present the
Q
-compensated viscoelastic reverse time migration (
Q
-ERTM) based on the decoupled fractional Laplacian (DFL) viscoelastic wave equation for high-resolution crosswell imaging. We develop an explicit stabilization scheme to resolve the cumbersome numerical instability issue in
Q
-ERTM. The merits of explicit stabilization are twofold. First, it simplifies the workflows of the
Q
-ERTM by avoiding domain transforms. In addition, it provides a flexible way for stabilization parameter tuning by introducing a reference scaling factor. We follow the best practices of high-performance computing with the MPI + CUDA configuration for numerical implementation. A toy crosswell imaging example and a more realistic time-lapse crosswell seismic survey with a CO
2
plume injection are provided to verify the feasibility and stability of the proposed method.
Attenuation compensation in prestack depth migration typically requires non-physical frequency-dependent energy amplification, which may lead to numerical instability. An explicit stabilization approach is developed for seismic Q compensation after deriving the Kspace Greens function of the compensated constant Q wave equation, which has decoupled fractional Laplacians (DFLs). At high wavenumbers, as time increases, the time propagator of K-space Greens function increases exponentially. Therefore, an exponential window function is introduced to stabilize the exponentially divergent time propagator. Unlike the conventional low-pass filtering approach in the frequency or wavenumber domain, the proposed method assumes that the exponent of the chosen window is a power function of the wavenumber magnitude, which only involves explicit stabilization terms in the timespace domain. An explicit stabilization form helps to perform seismic data Q compensation more conveniently. We outline the basic structure of the proposed approach with explicit stabilization and highlight some numerical details using CUDA-based implementations. The strong scaling analysis justifies the good performance of the developed code package in terms of computational efficiency and scalability. Besides, we further analyze the optimal scheme parameter selection and the influence of parameters on filtering performance. The proposed Q-RTM is applied on the Marmousi model and both synthetic and real crosswell examples to verify its feasibility and numerical stability.
High-precision seismic imaging is the core task of seismic exploration, guaranteeing the accuracy of geophysical and geological interpretation. With the development of seismic exploration, the targets become more and more complex. Imaging on complex media such as subsalt, small-scale, steeply dipping and surface topography structures brings a great challenge to imaging techniques. Therefore, the seismic imaging methods range from stacking- to migration- to inversion-based imaging, and the imaging accuracy is becoming increasingly high. This review paper includes: summarizing the development of the seismic imaging; overviewing the principles of three typical imaging methods, including common reflection surface (CRS) stack, migration-based Gaussian-beam migration (GBM) and reverse-time migration (RTM), and inversion-based least-squares reverse-time migration (LSRTM); analyzing the imaging capability of GBM, RTM and LSRTM to the special structures on three typical models and a land data set; outlooking the future perspectives of imaging methods. The main challenge of seismic imaging is to produce high-precision images for low-quality data, extremely deep reservoirs, and dual-complex structures.
Compared with PP-wave data, multicomponent seismic data have obvious advantages in describing complex geological structures during exploration. However, seismic energy is absorbed and dissipated by anelastic subsurface media during wave propagation, which makes seismic imaging more challenging, especially for PS-waves. In this article, we develop a
Q
-compensation method for both PP- and PS-waves based on the S-transform called
Q
-compensated prestack time migration (PSTM). Our migration approach has two key steps. First, we use a dip-angle model to optimize the migration aperture; consequently, the predominant diffraction energy from the Fresnel zones is located appropriately. Second, we use effective
Q
parameters to compensate for both PP- and PS-wave attenuation along the seismic wave propagation paths during migration. In addition, we propose some strategies for establishing the dip-angle model, estimating the effective
Q
values and matching PP- and PS-wave events. Furthermore, applications to synthetic and field data demonstrate that our migration method effectively recovers the amplitude and phase of seismic waves, so high-resolution migration sections are obtained.
Full waveform inversion (FWI) makes full use of the seismic waveforms to find high-resolution velocity and density models by minimising residuals between the calculated and recorded data. Q attenuation widely exists in the subsurface media, leading to weak amplitude and misplacement of reflectors. However, the commonly used Q-compensated FWI (QFWI) based on the second-order wave equation has difficulties in simultaneously inverting velocity and density fields. A QFWI method based on new first-order viscoacoustic quasi-differential equations is proposed to simultaneously produce velocity and density fields. Based on the adjoint state inversion theory, Q-compensated forward-propagated operators, adjoint operators, and gradient equations are derived using the newly derived first-order viscoacoustic quasi-differential wave equations. The time-domain multi-scale decomposition method is introduced to update the velocity and density models from a low to a high wavenumber. Numerical examples on an actual work area model and a modified attenuating Marmousi model show that the proposed QFWI method produces higher-accuracy velocity and density models with iterations by correcting the Q attenuation than the conventional acoustic FWI. Even when the Q model is extremely inaccurate, the proposed QFWI obtains acceptable inversion results. Compared to the conventional QFWI, our QFWI better inverts velocity field in the case of an inaccurate density model. Finally, we verify the adaptability of our QFWI to field data.
Elastic least‐squares reverse time migration can image the multi‐component seismic recordings. However, on the one hand, without considering the intrinsic attenuation of the subsurface, it may produce blurred reflectivity images with incorrect positions of reflectors for seismic recordings with strong attenuation. On the other hand, the crosstalk artifacts created by the different wave modes may severely degrade the imaging quality. To alleviate these crosstalk artifacts and compensate for the attenuation effects, we have developed a time‐domain wavefield‐separation‐based least‐squares reverse time migration approach in viscoelastic media, which utilizes the viscoelastic wave equation on the basis of the standard linear solid model to simulate intrinsic subsurface attenuation. The key point of the proposed approach is that we utilize the separated gradient contribution of PP and PS wave modes based on the wavefield separation technique in viscoelastic media to construct the P‐wave and S‐wave velocity images, respectively. Unlike the conventional wavefield‐separation‐based least‐squares reverse time migration approach which generally uses the new stress‐velocity equations to formulate, our proposed wavefield separation scheme fully depends on the conventional viscoelastic wave equation and its adjoint wave equation. By introducing the pure P‐wave stress in the forward and adjoint wavefields, the coupled wavefields can be well decomposed, which requires much less computational costs than the wavenumber‐domain wavefield separation scheme. Numerical examples using the layered and Marmousi‐2 models have shown that the proposed approach can improve the migration image quality under geological conditions with strong attenuation where elastic least‐squares reverse time migration may produce blurred and unfocused events. Meanwhile, the proposed least‐squares reverse time migration approach with the wavefield separation scheme has a better convergence rate and produces fewer crosstalk artifacts than that without the wavefield separation scheme. This article is protected by copyright. All rights reserved
We present Liouville partial-differential-equation (PDE) based methods for computing complex-valued eikonals in the multivalued (or multiple arrival) sense in attenuating media. Since the earth is comprised of attenuating materials, seismic waves usually attenuate so that seismic data processing calls for properly treating the resulting energy losses and phase distortions of wave propagation. In the regime of high-frequency asymptotics, the complex-valued eikonal is a crucial ingredient for describing wave propagation in attenuating media, since it is a unique quantity which summarizes two wave properties into one function: its real and imaginary parts are able to capture the effects of phase dispersions and amplitude attenuations, respectively. Because the usual ordinary-differential-equation (ODE) based ray-tracing methods for computing complex-valued eikonals distribute the eikonal solution irregularly in real space, we are motivated to develop PDE based Eulerian methods for computing complex-valued eikonals on regular meshes. Therefore, we propose to solve novel paraxial Liouville PDEs in real phase space so that we can compute the real and imaginary parts of the complex-valued eikonal in the multivalued sense on regular meshes. We dub the resulting method the Liouville PDE method for complex multivalued eikonals in attenuating media. We also provide Liouville PDE formulations for computing multi-valued amplitudes. Numerical examples, including a synthetic gas-cloud model, demonstrate that the proposed methods yield highly accurate complex-valued eikonals in the multivalued sense.
Intrinsic attenuation gives rise to phase dispersion and amplitude loss during seismic wave propagation. Not correcting these effects in seismic imaging can result in inaccurate reflector locations, dimmed amplitudes and degraded spatial resolution. In reverse-time migration (RTM), attenuation compensation can be implemented by reversing the sign of the dissipation term and keeping the dispersion term unchanged for backward wavefield extrapolation. Although this Q-compensated RTM scheme can effectively correct attenuation effects, amplitude amplification during back-propagation might lead to numerical instabilities, especially for field data with strong high-frequency noise. To mitigate this problem, we develop a robust space-wavenumber compensation operator, and apply it to viscoacoustic RTM. By analyzing the dispersion-only and viscoacoustic Green’s functions, we obtain an analytical solution for the attenuation compensation operator in a homogeneous medium. Because it is a time-frequency operator, to apply it directly in viscoacoustic RTM requires access to the extrapolated wavefields within a certain time window. To avoid storing the wavefields and improve computational efficiency, we use an approximated dispersion relation and convert the time-frequency operator to an equivalent space-wavenumber operator, which allows us to implement attenuation compensation on the fly during wavefield extrapolation. The hybrid-domain property of the operator enables us to account for the wavenumber-dependent compensation. A similar strategy can also be applied to the migrated images as a poststack processing approach, which is more efficient than the prestack compensation. Two synthetic and one land field dataset examples demonstrate the feasibility and adaptability of the proposed method.
Seismic wave attenuation caused by subsurface viscoelasticity reduces the quality of migration and the reliability of interpretation. A variety of Q-compensated migration methods have been developed based on the second-order viscoacoustic quasidifferential equations. However, these second-order wave-equation-based methods are difficult to handle with density perturbation and surface topography. In addition, the staggered grid scheme, which has an advantage over the collocated grid scheme because of its reduced numerical dispersion and enhanced stability, works in first-order wave-equation-based methods. We have developed a Q least-squares reverse time migration method based on the first-order viscoacoustic quasidifferential equations by deriving Q-compensated forward-propagated operators, Q-compensated adjoint operators, and Q-attenuated Born modeling operators. Besides, our method using curvilinear grids is available even when the attenuating medium has surface topography and can conduct Q-compensated migration with density perturbation. The results of numerical tests on two synthetic and a field data sets indicate that our method improves the imaging quality with iterations and produces better imaging results with clearer structures, higher signal-to-noise ratio, higher resolution, and more balanced amplitude by correcting the energy loss and phase distortion caused by Q attenuation. It also suppresses the scattering and diffracted noise caused by the surface topography.
Seismic waves in earth media usually undergo attenuation, causing energy losses and phase distortions. In the regime of high-frequency asymptotics, a complex-valued eikonal is an essential ingredient for describing wave propagation in attenuating media, where the real and imaginary parts of the eikonal function capture dispersion effects and amplitude attenuation of seismic waves, respectively. Conventionally, such a complex-valued eikonal is mainly computed either by tracing rays exactly in complex space or by tracing rays approximately in real space so that the resulting eikonal is distributed irregularly in real space. However, seismic data processing methods, such as prestack depth migration and tomography, usually require uniformly distributed complex-valued eikonals. Therefore, we propose a unified framework to Eulerianize several popular approximate real-space ray-tracing methods for complex-valued eikonals so that the real and imaginary parts of the eikonal function satisfy the classical real-space eikonal equation and a novel real-space advection equation, respectively, and we dub the resulting method the Eulerian partial-differential-equation method. We further develop highly efficient high-order methods to solve these two equations by using the factorization idea and the Lax-Friedrichs weighted essentially non-oscillatory (WENO) schemes. Numerical examples demonstrate that the proposed method yields highly accurate complex-valued eikonals, analogous to those from ray-tracing methods. The proposed methods can be useful for migration and tomography in attenuating media.
Pre-stack Q migration can eliminate the absorption effect and accurately image underground structures, which is conducive to subsequent reservoir interpretation and hydrocarbon prediction. However, the instability of Q migration amplifies high-frequency noise, which seriously reduces the imaging quality. To solve the instability problem, this paper studies the stability conditions for Q migration in the frequency domain. The generalised standard linear solid (GSLS) model can well describe the attenuation characteristics of underground media by combining different basic rheological models. Based on the Von Neumann stability analysis for the finite difference scheme combined with parameter settings in the GSLS model, this paper focuses on the stability of frequency domain Q migration and theoretically deduces the stability conditions suitable for the GSLS model. The given stability conditions can be directly implemented in the frequency domain Q migration process and constrain only the maximum reference angle frequency rather than the wave field frequencies, which avoids the Gibbs effect like the high-frequency cut method. In addition, the stability conditions can be adjusted adaptively with the computed frequencies, without the problem of over- or insufficient compensation. The model and practical application indicate that based on the GSLS model and its stability conditions, the attenuation effect can be compensated stably, lost energy and frequencies can be recovered, and high-quality imaging results are obtained.
Anisotropy and absorption are critical to the modeling and analysis of seismic amplitude,phase, and traveltime data. To neglect any of these phenomena, which are often bothoperating simultaneously, degrades the resolution and interpretability of migrated images.However, a full accounting of anisotropy and anelasticity is computationally complex andexpensive. One strategy for accommodating these aspects of wave propagation, while keepingcost and complexity under control, is to do so within an acoustic approximation. Weset up a procedure for solving the time-domain viscoacoustic wave equation for tilted transverselyisotropic (TTI) media, based on a standard linear solid model and, from this, developa viscoacoustic reverse time migration (Q-RTM) algorithm. In this approach, amplitudecompensation occurs within the migration process through a manipulation of attenuationand phase dispersion terms in the time domain differential equations. Specifically, theback-propagation operator is constructed by reversing the sign only of the amplitude lossoperators, but not the dispersion-related operators, a step made possible by reformulatingthe absorptive TTI equations such that the loss and dispersion operators appear separately.The scheme is tested on synthetic examples to examine the capacity of viscoacoustic RTM to correct for attenuation, and the overall stability of the procedure.
In the deep-marine environment, seismic data contain multiples and are seriously affected by
Q
attenuation. Multiples have been used in migration to image the shadow zones and improve the resolution. However, the effect of attenuation on multiples is more serious than primaries because multiples have longer propagation paths. Therefore, we compensate the forward-propagated source-side and backward-propagated receiver-side wavefields along all the propagation paths of multiples. To fully use the primaries and multiples, we construct an objective function of least-squares reverse time migration of joint primaries and multiples (LSRTM-J) to update the imaging results by jointly using primaries and different-order multiples. In practice in a seawater medium, seismic waves can hardly be affected by attenuation. To decrease the computational cost, we divide the medium in the deep-marine environment into an acoustic medium part and a viscoacoustic medium part and derive the acoustic–viscoacoustic coupled compensated forward continuation operator, compensated adjoint operator, attenuated demigration operator, and gradient formula of joint primaries and multiples. To eliminate the severe scattering and diffracted noise caused by the strong-reflected irregular seabed interface, we mesh the velocity and
Q
models into curvilinear grids to perfectly match the seabed structure and realize the proposed viscoacoustic LSRTM-J in the curvilinear domain. Numerical examples on two typical models and a real data test suggest that the proposed method produces images with high SNR, high resolution, balanced amplitude, clear imaging structures, and strong deep region energy, and the total computational cost is the least of the other four conventional methods.
It is widely known that strong noise can decrease the quality of seismic data. However, the anelastic attenuation could be more important to account for the weak amplitude and low quality of seismic data. Here, we develop an inversion framework to simultaneously compensate for the attenuation of seismic data and remove noise, thereby enhancing the quality of seismic data. Instead of directly applying a compensation operator to the input seismic data, we formulate an inverse problem that connects the sparse reflectivity model and the raw seismic data via the convolution and attenuation functions. The random noise is assumed to be the unpredicted part of the forward modeling process. We use the
L2
-norm regularization for the data misfit and impose a sparsity constraint onto the reflectivity series, e.g., using the
L1
-norm constraint. We use an iterative preconditioned conjugate gradient method to solve the
L1
-norm constrained least-squares optimization problem and obtain the reflectivity series. The denoised and compensated data are obtained by applying the convolution operator to the reflectivity. We use several synthetic and field seismic data to illustrate the effectiveness of the presented method.
Q-compensated reverse-time migration (Q-RTM) has been proven as an efficient method for seismic imaging with high fidelity. However, the source (forward) and receiver (backward) wavefields propagate along the opposite direction of time, and the recursive computation with the out-of-order access requires that all the wavefields of source propagation should be stored on the hard disk. For massive amounts of seismic data, saving the source wavefield from the central processing unit (CPU) [or graphics processing unit (GPU)] device to the disk and loading these data from the hard disk to the CPU (or GPU) device become extremely intensive in time and storage, which has been a bottleneck of Q-RTM. Several methods have been developed to reduce the huge wavefield storage in acoustic media, but are not applicable in the attenuated media. In this letter, we present a reversible hybrid absorbing boundary condition for Q-RTM, which is implemented by mixing the reversible attenuation and the random boundary conditions. Based on our developed new boundary, we just need to save the wavefield at the last one or two time steps in the forward process and then reconstruct the source wavefield in the time-reversal order. Numerical results demonstrate that the method can avoid the huge seismic data input and output (I/O) requirement and improve the computational efficiency dramatically.
Pre-stack reverse time migration (RTM) based on the two-way wave equation has been proved to be the most accurate seismic migration method theoretically. However, it requires reverse-order access to the wavefield calculated in forward time. In recursion computing, such out-of-order access requires that most of the recursion history should be stored on the hard disk. For massive amounts of seismic data, loading the saved wavefield data from the disk during imaging has been the bottleneck of RTM, restricting its wide application. To solve this problem, the wavefield in forward time must be reconstructed in reverse order. Although the random boundary can avoid the disk requirement by creating random velocity around the computational domain when propagate the source function. However, the random wavefield reflected from the boundary can generate unwanted artifacts in the final images. In this paper, we develop an attenuated and reversible random boundary condition which is implemented by mixing the reversible attenuation and random boundary conditions. Similar to the random boundary scheme, the proposed method just needs to save the last one or two wavefield snapshots into the memory in forward process. It then reconstructs the source wavefield in reverse order, while greatly reduces the disk input and output (I/O) requirements. Taking the attenuated property into consideration, the artificial events reflected from the boundary can be eliminated. Thus, our method can improve the imaging quality largely compared with the random boundary scheme. Numerical results demonstrate that the RTM images with our proposed attenuated and reversible random boundary condition can not only eliminate the unwanted artifacts, but also improve the computational efficiency greatly.
Spatial variations in the transmission properties of the overburden cause seismic amplitude attenuation, wavelet phase distortion and seismic resolution reduction on deeper horizons. This poses problems for the seismic interpretation, tying of migration images with well-log data and AVO analysis. We developed a prestack depth Q migration approach to compensate for the frequency dependent dissipation effects in the migration process. A 3D tomographic amplitude inversion approach may be used for the estimation of absorption model. Examples show that the method can mitigate these frequency dependent dissipation effects caused by transmission anomalies and should be considered as one of the processes for amplitude preserving processing that is important for AVO analysis when transmission anomalies are present.
In this paper, we compare the dissipation and dispersion characteristics of six visco-acoustic wave equations using numerical methods. We derive the expressions for spectral ratio and phase velocity for the six wave equations, which are used in the analysis. We then compute wave fields by numerically solving the six wave equations for the same model and compare their frequency spectrums, waveforms and quality factors, as calculated by the spectral ratio method. Finally, we conclude that four of the wave equations have analogous dissipation and dispersion characteristics, whereas the other two wave equations are quite different.
The anelastic properties of real materials, particularly of porous rocks, are described using the theory of linear viscoelasticity based on Boltzmann's superposition principle. Wave-propagation simulation with this model requires implementing the convolutional relation in the equation of motion. A pseudospectral time-integration technique is used to solve the equation of motion. Applications of viscoacoustic modeling suggest the need for considering the correct attenuation-dispersion effects for various fundamental seismic problems in anelastic earth models. -from Authors
Reliable analysis of amplitude variation with angle data requires that accurate seismic amplitude information be produced by prestack migration. Conventional prestack migration based on the scalar wave equation compensates for geometrical spreading, but not for transmission losses, intrinsic Q losses, or dispersion. Deterministic, model- and data-dependent corrections are performed as part of 2-D prestack migration that uses a viscoscalar, one-way, depth-stepping wave equation for extrapolation of both source and receiver wavefields in the frequency-space domain. Q compensation (for attenuation) is performed by including a Q-dependent term in the extrapolator. Dispersion is corrected using a frequency-dependent velocity model. The imaging condition is modified to provide a correction to the propagating source and receiver wavefields at each depth step to compensate for transmission losses. Tests use data from the Marmousi model. The final prestack imaged amplitudes produced by compensated prestack migration arc a close approximation to the correct angle-dependent reflection coefficients. There is only a small (~10%) increase in computation time over the traditional uncompensated migration.
Velocity dispersion is not usually a problem in surface seismic data processing, as the seismic bandwidth is relatively narrow and thus for most Q values, dispersive effects are not noticeable. However, for highly absorptive bodies, such as the overpressured free gas accumulations associated with some gas hydrates or high-porosity normally pressured gas sands, dispersive effects may be seen. In this work I analyse one such data set from the offshore north-east coast of India. I demonstrate that the effect is measurable and that compensating for it in either data processing or migration can improve the wavelet character, as well as delivering an estimate of the effective Q values in the associated geobody. I also raise the question as to whether velocities derived using low-frequency waveform inversion over such dispersive geobodies are wholly appropriate for migration of full seismic-bandwidth data.
We have developed a migration scheme that can compensate absorption and dispersion caused by intrinsic attenuation in subsurface media. The scheme was developed by adapting prestack time migration (PSTM) in the frequency domain. Instead of applying a commonly used Q factor, we devised an effective Q parameter to compensate absorption and dispersion. The effective Q determines the frequency-dependent traveltime and amplitude at one imaging location by only one value. As a result, the effective Q can be estimated by scanning technology. We designed an index that can remove the effects of interferences of the reflections resulting from stacked thin layers in extracting the effective Q parameter from scanning results. The proposed scheme can thus determine an effective Q model using surface seismic data during migration. Stabilization is achieved by introducing a smooth, maximum-limited gain function that matches the exact amplitude compensation factor when it is less than the user-specified gain limit. The proposed scheme can be incorporated into conventional PSTM workflow. Synthetic and field data sets were used to test the proposed deabsorption PSTM. Higher-resolution imaging results are obtained.
This book presents the different seismic data processing methods, also known as seismic "migration," in a unified mathematical way. The book serves as a bridge between the applied math and geophysics.
A 2-step plane-wave deconvolution scheme was demonstrated to be superior to conventional deconvolution procedures. Tests with field data indicate the method is effective in removing attenuation effects from both VSP (Vertical Seismic Profile) and surface measurements. Phase distortions are eliminated and interference between events is reduced within the seismic band. -from Authors
The amplitude dimming, frequency loss, and phase distortion of seismic images associated with shallow gas anomalies can be mitigated by using wave-equation migration algorithms that take the Earth's anelastic properties into account. Several field data examples from the Gulf of Mexico and West Africa illustrate improvements in seismic imaging and amplitude fidelity and demonstrate their impact on reservoir characterization and simulation.
Dispersive 1-D backward propagation can form the basis of a number of different algorithms for inverse Q filtering, each of which is akin to a particular migration algorithm. An especially efficient algorithm can be derived by means of a coordinate transformation equivalent to that in the Stolt frequency-wavenumber migration. This fast algorithm, valid for Q constant with depth, can be extended to accommodate depth-variable Q by cascading a series of constant Q compensations, as in cascaded migration. Data examples compare the results of conventional processing with the more stable phase treatment that can be obtained by including prestack inverse Q filtering in the processing. -from Authors
A linear model for attenuation of waves is presented, with Q, or the portion of energy lost during each cycle or wavelength, exactly independent of frequency. The wave propagation is completely specified by two parameters, e.g., Q and co, a phase velocity at an arbitary reference frequency wo. A simple exact derivation leads to an expression for the phase velocity c as a function of frequency: c/co = (w/wo), where gamma = (1/pi) tan-1 (1/Q). Scaling relationships for pulse propagation are derived and it is shown that for a material with a given value of Q, the risetime or the width of the pulse is exactly proportional to travel time. The travel time for a pulse resulting from a delta function source at x = 0 is proportioal to xbeta, where beta = 1/(1 - gamma). On the basis of this relation it is suggested that the velocity dispersion associated with anelasticity may be less ambiguously observed in the time domain than in the frequency domain. -Author
The concept of a relaxation spectrum is used to compute the absorption and dispersion of a linear anelastic solid. The Boltzmann after-effect equation is solved for a solid having a linear relationship between stress and strain and their first time derivatives, the ‘standard linear solid’, and having a distribution of relaxation times. The distribution function is chosen to give a nearly constant Q over the seismic frequency range. Both discrete and continuous relaxation spectra are considered. The resulting linear solid has a broad absorption band which can be interpreted in terms of a superposition of absorption peaks of individual relaxation mechanisms.
The accompanying phase and group velocity dispersion imply that one cannot directly compare body wave, surface wave, and free oscillation data or laboratory and seismic data without correcting for absorption. The necessary formalism for making these corrections is given. In the constant Q regions the correction is the same as that implied in the theories of Futterman, Lomnitz, Strick and Kolsky.
The Gulf of Mexico Gas Hydrates Joint Industry Project (JIP) is a consortium of production and service companies and some government agencies formed to address the challenges that gas hydrates pose for deepwater exploration and production. In partnership with the U.S. Department of Energy and with scientific assistance from the U.S. Geological Survey and academic partners, the JIP has focused on studies to assess hazards associated with drilling the fine-grained, hydrate-bearing sediments that dominate much of the shallow subseafloor in the deepwater (>500 m) Gulf of Mexico. In preparation for an initial drilling, logging, and coring program, the JIP sponsored a multi-year research effort that included: (a) the development of borehole stability models for hydrate-bearing sediments; (b) exhaustive laboratory measurements of the physical properties of hydrate-bearing sediments; (c) refinement of new techniques for processing industry-standard 3-D seismic data to constrain gas hydrate saturations; and (d) construction of instrumentation to measure the physical properties of sediment cores that had never been removed from in situ hydrostatic pressure conditions. Following review of potential drilling sites, the JIP launched a 35-day expedition in Spring 2005 to acquire well logs and sediment cores at sites in Atwater Valley lease blocks 13/14 and Keathley Canyon lease block 151 in the northern Gulf of Mexico minibasin province. The Keathley Canyon site has a bottom simulating reflection at ∼392 m below the seafloor, while the Atwater Valley location is characterized by seafloor mounds with an underlying upwarped seismic reflection consistent with upward fluid migration and possible shoaling of the base of the gas hydrate stability (BGHS). No gas hydrate was recovered at the drill sites, but logging data, and to some extent cores, suggest the occurrence of gas hydrate in inferred coarser-grained beds and fractures, particularly between 220 and 330 m below the seafloor at the Keathley Canyon site. This paper provides an overview of the results of the initial phases of the JIP work and introduces the 15 papers that make up this special volume on the scientific results related to the 2005 logging and drilling expedition.
Jacques LeveilleAttenuation compensation in viscoacoustic reverse time migration 3825-3830. [Abstract] [References] [PDF]
- Jianyong Bai
- Guoquan Chen
- David Yingst
Jianyong Bai, Guoquan Chen, David Yingst, Jacques LeveilleAttenuation compensation in viscoacoustic reverse time migration 3825-3830. [Abstract] [References] [PDF] [PDF w/Links] [Supplemental Material]
Zhiming RenComparisons of viscous acoustic wave equations 3365-3369. [Abstract] [References] [PDF]
- Zongqing Yang
- Yang Liu
Zongqing Yang, Yang Liu, Zhiming RenComparisons of viscous acoustic wave equations 3365-3369. [Abstract] [References] [PDF] [PDF w/Links] [Supplemental Material]
Brandsberg‐DahlWave equation migration with attenuation and anisotropy compensation 232-236 Redistribution subject to SEG license or copyright; see Terms of Use
- A A Valenciano
- N Chemingui
- D Whitmore
A. A. Valenciano, N. Chemingui, D. Whitmore, S. Brandsberg‐DahlWave equation migration with attenuation and anisotropy compensation 232-236. [Abstract] [References] [PDF] [PDF w/Links] Downloaded 02/27/15 to 169.230.243.252. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/