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True-amplitude prestack depth migration
Abstract
Most current true-amplitude migrations correct only for geometric spreading. We present a new prestack depth-migration method that uses the framework of reverse-time migration to compensate for geometric spreading, intrinsic Q losses, and transmission losses. Geometric spreading is implicitly compensated by full two-way wave propagation. Intrinsic Q losses are handled by including a Q-dependent term in the wave equation. Transmission losses are compensated based on an estimation of angle-dependent reflectivity using a two-pass recursive reverse-time prestack migration. The image condition used is the ratio of receiver/source wavefield amplitudes. Two-dimensional tests using synthetic data for a dipping-layer model and a salt model show that loss-compensating prestack depth migration can produce reliable angle-dependent reflection coefficients at the target. The reflection coefficient curves are fitted to give least-squares estimates of the velocity ratio at the target. The main new result is a procedure for transmission compensation when extrapolating the receiver wavefield. There are still a number of limitations (e.g., we use only scalar extrapolation for illustration), but these limitations are now better defined.
... Unfortunately, the amplitude loss and phase dispersion terms are coupled in the memory variable of these kind viscoacoustic wave equations. Causse and Ursin (2000) and Deng and McMechan (2007) reversed the sign of memory variables to compensate for the amplitude loss effects. However, the image energy is only partially compensated, and the interface mispositions still exist because the amplitude loss and phase dispersion are not fully decoupled (Guo et al., 2016). ...
... We illustrate the coupling of amplitude loss and phase dispersion terms in the memory variable. Subsequently, we analyse the reason behind the imperfect Q-compensated migration results obtained through the directly reverse memory variables (reverse-r) scheme (Deng & McMechan, 2007;Guo et al., 2016). Then, we derive a novel high-efficient decoupled viscoacoustic wave equation for wavefield simulation and migration in attenuating media. ...
... In fact, Causse and Ursin (2000), Deng and McMechan (2007) and Guo et al. (2016) simulated compensation wavefield in Q-RTM algorithm using the reverse-r scheme, rather than the above reverse imaginary part scheme. The dispersion relation controlling viscoacoustic waves of reverse-r scheme is as follows (the form in the time-domain is shown in Appendix A): ...
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 migration weights involved in true-amplitude Kirchhoff migration or the residual correction derived by Zhang et al. (2005b) are no longer explicit in RTM. To compensate the transmission losses, Zhang et al. (2003b) estimated reflection coefficients with a frequencyspace domain-migration method; Deng and McMechan (2007) present a prestack depth-migration method that uses the framework of reverse-time migration to compensate for geometric spreading, intrinsic Q losses, and transmission losses. In their method, they compensate the transmission losses for RTM with plane-wave transmission coefficients. ...
... In practice, a smooth model is often used for migration. Deng and McMechan (2007) proposed a more practical procedure to compensate transmission losses for the smooth model. Our derivations support the method of Deng and McMechan (2007) in theory. ...
... Deng and McMechan (2007) proposed a more practical procedure to compensate transmission losses for the smooth model. Our derivations support the method of Deng and McMechan (2007) in theory. ...
... The SLS model is the most realistic, consisting of a KV model connected in series with a spring. The relaxation function of this model is defined by (12) Thus, starting with Equation 1, using Equation 10, and following some steps, there are as follows: (13) Where ρ(x) is the density at position x, is the Bulk modulus, is the particle velocity vector, and is the source at position . The symbol represents a convolution operation, which describes the dissipation mechanism in a viscoacoustic medium in Equation 1. represents the magnitude of . ...
... illustrate the velocity and reflectivity, respectively. The density and Q factor were calculated by the Gardner relation [13] and empirical equation [8]. These models have discretized with nx = 995 and nz = 402 samples. ...
Disregarding the effect of seismic signal attenuation in gas and hydrocarbon zones leads to a final image with low resolution. However, a range of equations called viscoacoustics overcome such limitations. We use three secondorder equations based on Maxwell, Kelvin-Voigt (KV), and Standard Linear Solid (SLS) models. We analyze the dissipation and dispersion effects on each of them through seismograms. We also perform Reverse Time Migration (RTM) using the exact adjoint operators (Q-RTAM). All numerical experiments were implemented using Devito - a domain-specific language (DSL) and code generation framework to design highly optimized finite difference kernels for use in inversion methods.
... Therefore, the propagation-path-based compensation methods implemented during the seismic wave propagation have also been investigated by many researchers [23][24][25]. This category of compensation schemes includes Q-compensated ray-based migration [26], Q-compensated one-way wave equation-based migration (Q-OWEM) [27], and Q-RTM [28][29][30][31][32]. Among them, each one has its own advantages and disadvantages. ...
... Q-RTM can be performed either in the time domain [29] or in the frequency domain [43]. In this paper, we focus on investigating the FQ-RTM. ...
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.
... However, the propagated adjoint wavefield along the reversal time direction will be amplified in the Q-RTM approach, hence it requires a low-pass filter to stabilize the extrapolated adjoint wavefield with additional computational resources at each time step. Another approach utilizes the viscoacoustic wave equation based on the generalized standard linear solid (GSLS) theory to describe intrinsic subsurface attenuation and carries out the amplitude compensation by reversing the sign of the memory variable [8], [9], [10], [11], [12]. The main advantage of the Q-RTM approach using the GSLS wave equation is that it can utilize the local finite-difference (FD) operator to extrapolate the source and adjoint wavefields, which makes the Q-RTM approach graphical processing units (GPUs) friendly. ...
... where the variable with * denotes the adjoint wavefield. In the conventional Q-RTM approach based on the GSLS wave equation, Deng and McMechan [10] changed the sign of the memory variable in the receiver wavefield to implement the attenuation compensation at the expense of inaccurate phase velocity. However, the phase velocity of the adjoint wavefield in (14)- (18) is the same as that of the source wavefield. ...
Least-squares reverse time migration (LSRTM) has the potential to retrieve a high-resolution subsurface image. However, the standard acoustic LSRTM approach may produce a blurred image, if directly applying it to attenuated seismic recordings. In this letter, we developed a novel 3D Q-compensated image-domain LSRTM approach, denoted as Q-IDLSRTM. The Hessian matrix in the proposed approach is efficiently estimated from the point spread functions (PSFs) which are calculated by a combination of viscoacoustic Born modeling and reverse time migration (RTM) based on the generalized standard linear solid (GSLS) wave equation. The major advantage of the proposed image-domain inversion is that it is much faster than data-domain inversion. The L1 norm constraint and total variation (TV) regularization are used to produce a sparse solution and maintain the structural continuity of the inverted image. We determine the effectiveness of the proposed approach with a part of the 3D Overthrust model and the resulting images demonstrate the ability of our approach to image subsurface structures with enhanced resolution and balanced amplitude relative to the RTM image and inverted image from the acoustic image-domain LSRTM approach.
... For Voigt-Ricker model, the attenuation effect is regarded as an energy absorption term, which is described by adding a single damping term to the acoustic wave equation in the model (Ricker, 1953). Because of this simplicity, the theory of wave propagation with absorption has been widely adopted to compute the synthetic seismograms in early researches (Boore et al., 1971;Munasinghe and Farnell, 1973;Deng and McMechan, 2007). However, the Q varies with frequency. ...
... First, based on the second-order acoustic wave equation, a first-order time derivative of pressure wavefields describes the energy dissipation (Ricker, 1953). Because of this simplicity, this wave equation theory with absorption has been widely adopted to compute the synthetic seismograms in early researches (Boore et al., 1971;Munasinghe and Farnell, 1973;Deng and McMechan, 2007). However, the Q varies with frequency, which contradicts observations. ...
Seismic attenuation is a useful physical parameter to describe and to image the properties of specific geological bodies, e.g., saturated rocks and gas clouds. Classical approaches consist of analyzing seismic spectrum amplitudes or spectrum distortions based on ray methods. Full waveform inversion is an alternative approach that takes into account the finite frequency aspect of seismic waves. In practice, both seismic velocities and attenuation have to be determined. It is known that the multi-parameter inversion suffers from cross-talks.This thesis focuses on retrieving velocity and attenuation. Attenuation dispersion leads to equivalent kinematic velocity models, as different combinations of velocity and attenuation have the same kinematic effects. I propose a hybrid inversion strategy: the kinematic relationship is a way to guide the non-linear full waveform inversion. The hybrid inversion strategy includes two steps. It first updates the kinematic velocity, and then retrieves the velocity and attenuation models for a fixed kinematic velocity. The different approaches are discussed through applications on 2D synthetic data sets, including the Midlle-East and Marmousi models.
... In the context of Q-RTM algorithms, there exist two variations of viscoacoustic wave equations: one involving the coupling of amplitude attenuation and phase dispersion terms, and the other with these terms decoupled. Drawing from the standard linear solid (SLS) model, references [6,7] derived a viscoacoustic wave equation that couples amplitude attenuation and phase dispersion, laying the foundation for the Q-RTM algorithm. However, in Q-RTM, the authentic procedure entails altering the sign of the amplitude attenuation term while retaining the sign of the phase dispersion term [8,9]. ...
Q-compensated reverse time migration (Q-RTM) is a crucial technique in seismic imaging. However, stability is a prominent concern due to the exponential increase in high-frequency ambient noise during seismic wavefield propagation. The two primary strategies for mitigating instability in Q-RTM are regularization and low-pass filtering. Q-RTM instability can be addressed through regularization. However, determining the appropriate regularization parameters is often an experimental process, leading to challenges in accurately recovering the wavefield. Another approach to control instability is low-pass filtering. Nevertheless, selecting the cutoff frequency for different Q values is a complex task. In situations with low signal-to-noise ratios (SNRs) in seismic data, using low-pass filtering can make Q-RTM highly unstable. The need for a small cutoff frequency for stability can result in a significant loss of high-frequency signals. In this study, we propose a multi-task learning (MTL) framework that leverages data-driven concepts to address the issue of amplitude attenuation in seismic records, particularly when dealing with instability during the Q-RTM (reverse time migration with Q-attenuation) process. Our innovative framework is executed using a convolutional neural network. This network has the capability to both predict and compensate for the missing high-frequency components caused by Q-effects while simultaneously reconstructing the low-frequency information present in seismograms. This approach helps mitigate overwhelming instability phenomena and enhances the overall generalization capacity of the model. Numerical examples demonstrate that our Q-RTM results closely align with the reference images, indicating the effectiveness of our proposed MTL frequency-extension method. This method effectively compensates for the attenuation of high-frequency signals and mitigates the instability issues associated with the traditional Q-RTM process.
... In this respect, (1) and (2) appear to be more mathematically equivalent to the viscoacoustic wave equation instead of the standard wave equation [55] v 2 ...
Diffraction separation is considered of great importance for the full-wavefront reconstruction (FWR) of ground penetrating radar (GPR) and its delicate data processing. In contrast to reflections, GPR diffraction events encode the fully dynamic feature and subwavelength information to identify small-scale heterogeneities, thereby offering enhanced illumination and resolution for GPR imaging. However, fully extracting these faint diffraction components remains a very challenging task. These components are usually ignored because of the strong interference and masking effect of the more dominant reflective components, and in particular, the attenuation of GPR waves accelerates almost exponentially with the frequency. In this article, we developed a coherence analysis (CA) framework for extracting GPR diffractions by considering the kinematic attributes and discontinuity features of GPR wavefronts, as well as the dispersion and attenuation effects of GPR responses. To be specific, the compensation preprocessing and semblance summation are performed sequentially for coherence acquisition, after which a multiscale subtraction is designed toward the coherent fields. In particular, we concentrated on analyzing media losses and multiparameter impacts on diffraction separation performance. To demonstrate the feasibility and practicality of the proposed framework, we tested our algorithm using synthetic and field datasets, both results with quantified metrics showing the elimination of continuities. The comparative results reveal the performance of the framework in preserving the edge components of GPR diffractions, making it potentially attractive for microscopic velocity analysis, refined full wavefield migration (FWM), and well-constrained full waveform inversion in terms of small-scale heterogeneities and discontinuities.
... At first, Causse and Ursin (2000) use a Q-RTM method to correct the absorption effects. Using the visco-acoustic WE for the standard linear solid (SLS) model, Deng and Mcmechan (2007) implement true-amplitude RTM to compensate for transmission losses, absorption attenuation, and geometric spreading. After that, Deng and Mcmechan (2008) further extend their method into visco-elastic media. ...
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.
... The main benefit of the RTM is that it migrates the entire wavefield, even commonly neglected contributions like multiples. In addition to that, RTM gained popularity, because it preserves amplitude information if propagation related amplitude losses are compensated (Deng and McMechan (2007), Chattopadhyay and McMechan (2008)) and because of its ability to handle signals from steep structures, which challenge the commonly applied Kirchhoff migration (Chattopadhyay and McMechan (2008)). A downside to RTM is that, even with an ideal seismic velocity model, numerical artefacts can form during RTM when ray paths are asymmetrical, for instance due to strong heterogeneity, which challenges both the Pre-and Post-Stack imaging criterion. ...
The inclusion of diffractions into inversion methods is suggested to widely complement
conventional seismic routines, because of the diffraction’s superior illumination capacities and sub-wavelength resolution. Nevertheless, diffraction focussed inversion methods are still chronically underrepresented, likely due to the diffraction’s weak relative
amplitudes (i.e bad visibility) and because many commonly applied inversion tools
are developed solely for reflections and refractions. However, the advance of inversion
techniques based on the Finite Differences (FD) solution of the wave equation, such as
Full waveform inversion (FWI) simplifies the inversion of diffractions, since said techniques can describe diffractions as well as reflections and refractions. In this study we developed and tested a simple 4 step diffraction focussed inversion approach. The first step is the determination of the kinematic wavefront attributes (kin. wfA), which are used in the second step to identify the diffractions and increase their visibility in the Post-Stack data. Thereafter, the data is sorted back into the Pre-Stack domain and exported as shotfiles once with the diffraction enhancing modifications and once without. The 3. step comprises the FWI of the entire, unmodified wavefield, the result of which is then used as initial model for the final step, a FWI of the diffraction enhanced data set. The developed diffraction focussed FWI approach was tested on two synthetic data sets, which while validating the approach’s potential to correctly identify and enhance diffractions, yielded inconclusive results in regard of the benefits of the diffraction focussed FWI (i.e., step 4). Though, because a field data application displayed significant improvements in the imaging capacities of Reverse Time Migration (RTM) as well as, seemingly, in the FWI resolution, the developed method is, nevertheless seen as promising. However, in order to decisively evaluate the method, further, better adapted synthetic
data tests are suggested.
... The early time-domain simulation approaches generally describe Q behavior with an approximate constant-Q (ACQ) model, which is built by superimposing several standard linear solid (SLS) elements in parallel over a specific frequency range (Liu et al., 1976;Day and Minster, 1984;Emmerich and Korn, 1987;Carcione et al., 1988;Zhu et al., 2013). Many SLS-based viscoacoustic wave equations are proposed and used as forwarding engines for attenuation compensated RTM, least-squares reverse time migration (LSRTM), and full waveform inversion (FWI) (Deng and McMechan, 2007;Dutta and Schuster, 2014;Bai et al., 2014;Li et al., 2019). Due to their computational efficiency and flexibility for domain decomposition in large-scale parallel computation, the viscoacoustic wave equations based on the ACQ model have become one of the most widely used modeling approaches in the petroleum industry (Hu et al., 2016;Zhou et al., 2018;Zhou et al., 2020). ...
The pseudo-viscoacoustic anisotropic wave equation is widely used in the oil and gas industry for modeling wavefields in attenuating anisotropic media. Compared with the full viscoelastic anisotropic wave equation, it can greatly reduce the computational cost of wavefield modeling while maintaining the visco-qP-wave kinematics very well. However, even if we place the source in a thin isotropic layer, there will be some unwanted shear wave artifacts in the qP-wave field simulated by the pseudo-viscoacoustic anisotropic wave equation due to the stepped approximation of inclined layer interfaces. Furthermore, the wavefield simulated by the pseudo-viscoacoustic anisotropic wave equation may suffer from numerical instabilities when the anisotropy parameter epsilon is less than delta. To overcome these problems, we derive a pure-viscoacoustic TTI wave equation in media with anisotropy in velocity and attenuation based on the exact complex-valued phase velocity formula. The pure-viscoacoustic TTI wave equation has decoupled amplitude dissipation and phase dispersion terms, which is suitable for further reverse time migration with Q-compensation. For numerical simulations, we adopt the second-order Taylor series expansion to replace the mixed-domain spatially variable fractional Laplacian operator, which guarantees the decoupling of the wavenumber from the space-related fractional order. Then, we use an efficient and stable hybrid finite-difference and pseudo-spectral method to solve the pure-viscoacoustic TTI wave equation. Numerical tests indicate that the simulation results of the newly derived pure-viscoacoustic TTI wave equation are stable, free from shear wave artifacts, and accurate. We further demonstrate that hybrid finite-difference and pseudo-spectral method outperforms pseudo-spectral method in terms of numerical simulation stability and computing efficiency.
... Reverse-time migration, as a representative wave-equation migration approach, has the potential to accurately image steep discontinuities and complex geological structures (Chang WF and McMechan, 1994;Du QZ et al., 2012, 2017Liu YS et al, 2019). Various studies have incorporated Q-compensation into reverse-time migration, with the Kjartansson constant Q theory (Carcione et al., 2002;Treeby and Cox, 2010;Zhang Y et al., 2010;Zhu TY et al., 2014;Sun JZ et al., 2015) and the generalized standard linear solid theory (Carcione et al., 1988;Robertsson et al., 1994;Blanch et al., 1995;Deng F and McMechan, 2007) being the two primary mechanisms used for Q parameterization (Guo P et al., 2016). Q-compensation has also been investigated in Gaussian beam migration (Bai M et al., 2016;Shi XC et al., 2019), which provides a competitive alternative that balances the computational costs with the precision of migration. ...
Because of the viscoelasticity of the subsurface medium, seismic waves will inherently attenuate during propagation, which lowers the resolution of the acquired seismic records. Inverse-Q filtering, as a typical approach to compensating for seismic attenuation, can efficiently recover high-resolution seismic data from attenuation. Whereas most efforts are focused on compensating for high-frequency energy and improving the stability of amplitude compensation by inverse-Q filtering, low-frequency leakage may occur as the high-frequency component is boosted. In this article, we propose a compensation scheme that promotes the preservation of low-frequency energy in the seismic data. We constructed an adaptive shaping operator based on spectral-shaping regularization by tailoring the frequency spectra of the seismic data. We then performed inverse-Q filtering in an inversion scheme. This data-driven shaping operator can regularize and balance the spectral-energy distribution for the compensated records and can maintain the low-frequency ratio by constraining the overcompensation for high-frequency energy. Synthetic tests and applications on prestack common-reflection-point gathers indicated that the proposed method can preserve the relative energy of low-frequency components while fulfilling stable high-frequency compensation.
... Geophysicists use parallel of relaxation mechanisms (e.g., Maxwell, Kelvin-Voigt and standard linear solid [SLS] mechanical model) to construct NCQ model in a given frequency band (e.g., Zener, 1948;Emmerich and Korn, 1987;Day, 1998;Carcione et al., 2004;Carcione, 2007;Cao and Yin, 2014). The viscoacoustic and viscoelastic wave equations based on NCQ model have been proposed and widely used for migration imaging (e.g., Carcione, 1988;Carcione et al., 1988;Liao and Mcmechan, 1993;Deng and McMechan, 2007;Dutta and Schuster, 2014;Qu et al., 2017;Li et al., 2019). However, as for viscoacoustic and viscoelastic wavefield simulations using multiple relaxation mechanisms, the complexity of gradient derivation will be greatly increased, which brings difficulties to the implementation of least-squares reverse time migration (LSRTM) and full-waveform inversion (FWI, Askan et al., 2007). ...
Time-domain constant-Q (CQ) viscoelastic wave equations have been derived to efficiently model Q, but are known to break down in accuracy in describing CQ attenuation at low Q. In view of this, a new time-domain viscoelastic wave equation for modeling wave propagation in anelastic medium is evaluated based on Kjartansson’s CQ model to improve the accuracy in describing CQ attenuation at low Q. We use an approximate frequency-domain viscoelastic wave equation to replace the accurate frequency-domain viscoelastic wave equation. Then, a new time-domain wave equation is derived by converting the approximate viscoelastic wave equation from the frequency domain to the time domain. The newly derived viscoelastic wave equation consists of several Laplacian differential operators with variable fractional order. We use an arbitrary-order Taylor series expansion (TSE) to approximate the derived mixed domain fractional Laplacian operators, and realize the decoupling of the wavenumber and fractional order. Then, the proposed viscoelastic wave equation can be solved directly using the staggered-grid pseudospectral method (SGPSM). We evaluate the precision of the new viscoelastic wave equation by comparing the numerical solutions with the analytical solutions in homogeneous medium. Theoretical curve analysis and numerical results indicate that the proposed fractional viscoelastic wave equation has higher precision in describing CQ attenuation than that of the traditional fractional viscoelastic wave equation, especially for cases that P-wave quality factor QP is less than 10, and S-wave quality factor QS is less than 8. Furthermore, we use two numerical examples to verify the effectiveness of the TSE SGPSM in heterogeneous media. The discussion shows that the advantage of using our fractional viscoelastic wave equation over the traditional fractional viscoelastic wave equation is the higher precision in describing CQ attenuation at different frequency.
... Under this theory, many ACQ viscoacoustic wave equations are derived in the time domain (Day and Minster, 1984;Emmerich and Korn, 1987;Carcione et al., 1988;Zhu et al., 2013). To date, these ACQ wave equations are still widely used for RTM, least-squares RTM, and full-waveform inversion (FWI) because of their high efficiency in numerical simulation (Deng and McMechan, 2007;Bai et al., 2014;Dutta and Schuster, 2014;Qu et al., 2017). The phase dispersion and amplitude loss effects of the preceding ACQ wave equations are coupled together, which makes it difficult to correct the phase dispersion and compensate the amplitude attenuation simultaneously in implementing Q-compensated RTM (Yang and Zhu, 2018). ...
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.
... Reverse time migration (RTM) is a seismic imaging method with high imaging accuracy (Whitmore 1983;McMechan 1983;Baysal et al. 1983;Levin 1984;Chang & McMechan 1986Deng & McMechan 2007;Li et al. 2019). Based on the two-way wave equation and primary reflections, it consists of three main steps: (1) seismic extrapolation along time using the estimated seismic wavelet as the source (shot-side wavefield), (2) inverse time wavefield extrapolation taking the actual seismic record acquired by geophones as the source (geophone-side wavefield), and (3) the application of imaging conditions (Sava & Fomel 2006;Chattopadhyay & McMechan 2008;Xu et al. 2010;Yu et al. 2011;Chung et al. 2012;Zhang et al. 2015). ...
Imaging of vertical structures is a challenge in the seismic imaging field. The conventional imaging methods for vertical structures are highly dependent on the reference model or boreholes. Time-reversed mirror imaging can effectively image the vertical structures based on the multiples and a smoothed velocity model without the need of accurate seismic wavelet estimation. Although the Laplacian operator is applied in time-reversed mirror imaging, there still exists severe residual noise. In this study, we developed a new imaging denoising strategy and an X-shaped supplement denoising operator for time-reversed mirror imaging based on the geometric features of the image and the causes of imaging noise. Synthetic results for the single- and double-staircase model prove the powerful denoising capacity of the X-shaped supplement denoising operator. In addition, the results of a Marmousi model prove that the X-shaped denoising operator can also effectively suppress the noise when applying time-reversed mirror imaging method to image complex inclined structures. However, the X-shaped denoising operator still contains some limitations, such as non-amplitude-preserving.
... 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.
... For Q-RTM algorithm, there are two kinds of viscoacoustic wave equations, i.e., the amplitude attenuation term and phase dispersion term coupled or decoupled equations. Based on the standard linear solid model (SLS), Deng and McMechan derived an amplitude attenuation and phase dispersion coupled viscoacoustic wave equation, and implemented the Q-RTM algorithm [6,7]. However, in Q-RTM, the true algorithm is to change the sign of the amplitude attenuation term, and keep the sign of the phase dispersion term [31]. ...
... The seismic quality factor (Q) is usually used to describe seismic wave attenuation, and it has been found to be nearly independent of the frequency within the exploration seismic frequency band (McDonal et al., 1958). The frequency-independent attenuation model is referred to as the constant-Q (CQ) model (Kjartansson, 1979), which has been widely used in Q-compensated reverse time migration (Q-RTM) (Deng and McMechan, 2007;Zhang et al., 2010;Dutta and Schuster, 2014;Sun et al., 2015) and full-waveform inversion (FWI) (Malinowski et al., 2011;Kurzmann et al., 2013;Bai et al., 2014). ...
We have developed a matrix-transform method (MTM) to numerically solve the fractional Laplacian constant-[Formula: see text] viscoacoustic wave equation. The new method is based on a matrix representation of the fractional Laplacians, and it is different from the traditional Fourier-spectrum representation. With the MTM approach, the viscoacoustic wave equation is discretized in the space domain; thus, the periodic boundary caused by the Fourier-transform method can be avoided. Spatial discretization offers great convenience for handling various boundary conditions. In the MTM scheme, the fractional power of a matrix needs to be computed, and it can be realized with the help of eigenvalue decomposition (EVD). However, the application of EVD to a huge matrix is uneconomical and impractical. To avoid forming any huge matrix explicitly, we adopted the Lanczos approximation to compute the matrix-vector product directly. The key step in the Lanczos approximation is the involved Lanczos decomposition, which is implemented by an iterative flow. We determine how to choose the iteration number in the Lanczos decomposition to balance the wavefield simulation accuracy and the computational cost with the help of numerical examples. We also analyze the stability condition of MTM numerically, and we compare its computational cost with that of the traditional Fourier-transform method.
... For Q-RTM algorithm, there are two kinds of viscoacoustic wave equations, i.e., amplitude attenuation and phase dispersion terms coupled or decoupled equations. Based on the standard linear solid model (SLS), Deng and McMechan [2] and Deng [3] derived an amplitude attenuation and phase dispersion coupled viscoacoustic wave equation and implemented the Q-RTM algorithm. However, in Q-RTM, the true algorithm aims to change the sign of the amplitude attenuation term, while maintaining the sign of the phase dispersion term [4]. ...
italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q
-compensated reverse time migration (
Q
-RTM) can compensate seismic attenuation caused by the anelastic behavior of subsurface media. Although, the traditional
Q
-RTM has high computational efficiency, it is instable because the high frequency or wavenumber ambient noise is exponentially boosted during forward and backward seismic wavefield propagation. The existing stable
Q
-RTM method costs twice as much computing time and memory compared to the traditional
Q
-RTM. In this letter, we propose a new
Q
-RTM method to address the above issues simultaneously. First, a new viscoacoustic wave equation is derived based on a wavefield decomposition method to obtain the velocity-dispersion-only and viscoacoustic wavefields efficiently. Then, a theoretical framework of stable and high-efficiency
Q
-RTM method is proposed based on the velocity-dispersion-only and viscoacoustic wavefields. The synthetic example shows that the new stable
Q
-RTM results match well with the reference images (without attenuation images). Moreover, the field data images also demonstrate the stability and high-efficiency of our proposed
Q
-RTM method.
The anisotropy and attenuation properties of real earth media can lead to amplitude reduction and phase dispersion as seismic waves propagate through it. Ignoring these effects will degrade the resolution of seismic imaging profiles, thereby affecting the accuracy of geological interpretation. To characterize the impacts of viscosity and anisotropy, we formulate a modified pure-viscoacoustic (PU-V) wave equation including the decoupled fractional Laplacian (DFL) for tilted transversely isotropic (TTI) media, which enables the generation of stable wavefields that are resilient to noise interference. Numerical tests show that the newly derived PU-V wave equation is capable of accurately simulating the viscoacoustic wavefields in anisotropic media with strong attenuation. Building on our TTI PU-V wave equation, we implement stable reverse time migration technique with attenuation compensation (Q-TTI RTM), effectively migrating the impacts of anisotropy and compensates for attenuation. In the Q-TTI RTM workflow, to remove the unstable high-frequency components in attenuation-compensated wavefields, we construct a stable attenuation-compensated wavefield modeling (ACWM) operator. The proposed stable ACWM operator consists of velocity anisotropic and attenuation anisotropic parameters, effectively suppressing the high-frequency artifacts in the attenuation-compensated wavefield. Synthetic examples demonstrate that our stable Q-TTI RTM technique can simultaneously and accurately correct for the influences of anisotropy and attenuation, resulting in the high-quality imaging results.
Visco-acoustic full waveform inversion (FWI) is a widely-used high-resolution seismic inversion method. It aims to achieve a joint inversion of the velocity and quality factor (
Q
) models. The resolution of inversion results highly depends on the accuracy of the seismic wave simulation. In this paper, the symplectic stereo-modeling method (SSM) is used to solve the visco-acoustic wave equation. We compare a series of numerical properties between SSM and the traditional finite difference (FD) methods like Lax-Wendroff correction (LWC) method, including numerical dispersion, accuracy, efficiency, numerical errors, stability, etc. The results show that the maximum numerical dispersion error of SSM is about 9%, while that of LWC is about 24%. Meanwhile, SSM is closer to the analytical solution, computationally efficient and stable. We derive the velocity and
Q
gradients for visco-acoustic FWI based on the adjoint-state method using the SSM method, namely SSM-based FWI. For the visco-acoustic Marmousi model, the results show that the proposed method has high inversion accuracy and minor numerical dispersion. Furthermore, for the field data, we implement a three-stage FWI for different frequency ranges and demonstrate the effectiveness of the proposed method.
Since the appearance of wave-equation migration, many have tried to improve the resolution and effectiveness of this technology. Least-squares wave-equation migration is one of those attempts that tries to fill the gap between the migration assumptions and reality in an iterative manner. However, these iterations do not come cheap. A proven solution to limit the number of least-squares iterations is to correct the gradient direction within each iteration via the action of a preconditioner that approximates the Hessian inverse. However, the Hessian computation, or even the Hessian approximation computation, in large-scale seismic imaging problems involves an expensive computational bottleneck, making it unfeasible. Therefore, we propose an efficient computation of the Hessian approximation operator, in the context of one-way wave-equation migration (WEM) in the space-frequency domain. We build the Hessian approximation operator depth by depth, considerably reducing the operator size each time it is calculated. We prove the validity of our proposed method with two numerical examples. We then extend our proposal to the framework of full-wavefield migration, which is based on WEM principles but includes interbed multiples. Finally, this efficient preconditioned least-squares full-wavefield migration is successfully applied to a dataset with strong interbed multiple scattering.
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.
Multiples have longer propagation paths and smaller reflection angles than primaries for the same source‐receiver combination, so they cover larger illumination area. Therefore, multiples can be used to image shadow zones of primaries. Least‐squares reverse time migration of multiples can produce high quality images with fewer artefacts, high resolution and balanced amplitudes. However, viscoelasticity exists widely in the earth, especially in the deep‐sea environment, and the influence of Q attenuation on multiples is much more serious than primaries due to multiples have longer paths. To compensate for Q attenuation of multiples, Q‐compensated least‐squares reverse time migration of different‐order multiples is proposed by deriving viscoacoustic Born modeling operators, adjoint operators and demigration operators for different‐order multiples. Based on inversion theory, this method compensates for Q attenuation along all the propagation paths of multiples. Examples on a simple four‐layer model, a modified attenuating Sigsbee2B model and a field data set suggest that the proposed method can produce better imaging result than Q‐compensated least‐squares reverse time migration of primaries and regular least‐squares reverse time migration of multiples. This article is protected by copyright. All rights reserved
The coastal structures, such as breakwaters and reefs, are usually vulnerable to the strong earthquake waves propagating through their bedrock foundation. This study proposes a 2D non-water substructure model in time domain for breakwater-water-bedrock (layered) (BWB) system, where the fluid-solid dynamic interaction is embodied by added mass and the equivalent seismic loading calculated from ocean free-field (without structure) accomplishes the inclined incidence of P or SV wave with any angle. Besides, the cutoff boundaries of the fluid are applied exact absorbing boundary conditions (ABCs) to simulate the far-field of water while the viscous-spring ABCs are employed to simulate the far-field of bedrock. Then, the accuracy of substructure method is verified by some numerical examples. However, the exact added mass matrix is fully populated and not easy to use for general software. Therefore, the simplified added mass models for water-bedrock system, water-structure system and BWB system are furtherly developed, including the relevant simplified formulas fitted by curve fitting method. Additionally, the numerical results indicate the good effectiveness of proposed model.
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.
In 2D anisotropic media, non-stationary filters and low-rank approximation methods are classical strategies to compute the decomposition operators, but they suffer from expensive computational costs for 3D media. This study adopts the eigenform analysis into 3D vertical transverse isotropic (VTI) media and produces the separated vector P and S wavefields with the same amplitudes, phases, and physical units as the input elastic wavefields. We first built a 3D zero-order pseudo-Helmholtz decomposition operator by deriving the eigenvalues and eigenvectors of the 3D VTI wave equations in the wavenumber domain. The eigenvalues refer to the phase velocities of P-, SH-, and SV-wave, and the corresponding eigenvectors are pointing to their polarizations. Second, we use the pseudo-Helmholtz decomposition operator to construct a 3D anisotropic Poisson’s equation. Based on the laterally homogeneous assumption, Poisson’s equation is solved in the mixed domain , where , , and z denote the horizontal wavenumbers and depth, respectively. Third, we obtain the vector P and S wavefields using the proposed 3D pseudo-Helmholtz decomposition operator in the space domain. Lastly, 3D PP and PS images are calculated with a dot-product imaging condition. The anisotropic amplitude versus offset (AVO) responses of the 3D elastic reverse-time migration (ERTM) images are also validated by analytical solutions (Ruger’s equations). Our proposed 3D pseudo-Helmholtz decomposition operator degrades to a gradient operator satisfying isotropic media conditions. In addition, the method is easy to extend into 3D due to its high-efficiency cost. Several numerical examples with large shear anisotropy are selected to demonstrate the feasibility of our proposed pseudo-Helmholtz decomposition method in 3D applications.
A numerical model for the structure-water-soil (layered)-rock (SWLR) system is proposed in this paper to evaluate the system dynamic response, in which the fluid-solid coupling effect is considered by the interface conditions and the wave radiation effects of far-field domain are simulated by artificial boundary conditions. Moreover, this model covers four prime respects, including the discretization of interface conditions, the absorbing boundary conditions, the equivalent earthquake loading input and the general finite element equation. It can be solved by a standard time integration algorithm such as implicit Newmark's method. Furtherly, the effectiveness of the proposed model is verified by three numerical examples. Apart from these, the dynamic stiffness matrix method (DSM) of fluid and solid adopted in present study to acquire the equivalent earthquake loading effectively accomplishes the oblique incidence of the wave in layered half-space with arbitrary angles. Based on the proposed model, the effects of dynamic water-structure interaction (WSI) and soil-water interaction (SWI) on the seismic responses of offshore structure are investigated, which is conducive to the simplified analysis of dynamic fluid-solid interaction and the design of ocean engineering.
Prismatic waves carry steeply dipping structural information that primaries cannot contain. Therefore, prismatic waves are separately used in some migration methods to improve the illumination and imaging effect on steeply dipping structures. Least-squares reverse time migration of prismatic waves (LSRTM-P) can produce high-resolution images with improved steeply dipping structures. However, viscoelasticity exists widely on the Earth, which poses great difficulty for imaging. The effect of attenuation on prismatic waves is difficult to be compensated when conducting LSRTM-P because prismatic waves have three propagation paths. To overcome this problem, a
Q
-compensated LSRTM (
Q
-LSRTM)-P method is proposed by deriving
Q
-compensated forward-propagated operators and backward-propagated adjoint operators of prismatic waves, which compensates for
Q
attenuation along all the three propagation paths of prismatic waves. The proposed
Q
-LSRTM-P is conducted to update the image after applying the conventional
Q
-LSRTM. Besides, the proposed method can be adapted to the irregular surface media. Numerical examples on two synthetic and a field datasets verify that our method can produce better imaging results with clearer steeply dipping structures, higher signal-to-noise ratio (SNR), higher resolution, and more balanced amplitude than noncompensated LSRTM-P and conventional
Q
-LSRTM.
This article presents a hybrid 3-D electromagnetic (EM) full-wave inversion (FWI) method for the reconstruction of subsurface objects illuminated by an antenna array with the limited aperture. The 3-D linear sampling method (LSM) is first used to qualitatively reconstruct the rough shapes and locations of the subsurface objects. Then, the 3-D convolutional neural network (CNN) U-Net is used to further refine the images of the unknown objects. Finally, the Born iterative method (BIM) is implemented to quantitatively invert for the dielectric parameters of subsurface inhomogeneous objects or multiple homogeneous objects in the restricted image regions. Numerical simulations show that, compared with the pure FWI method BIM, the proposed hybrid method can reconstruct subsurface 3-D objects from limited-aperture EM data with both higher accuracy and lower computational cost. In addition, the proposed hybrid method also shows a strong antinoise ability for the reconstruction of multiple subsurface objects.
Natural gas hydrate (NGH) is a potential clean alternative energy source for fossil fuels. In seismic imaging profiles, NGH is often identified by the bottom-simulating reflection (BSR), which is characterized by strong reflection amplitude and negative polarity. High-resolution and amplitude-preserved seismic imaging are demanded for the detection of NGH. However, the traditional acoustic reverse-time migration ignores the attenuation characteristics of the medium, which leads to reduced amplitude and distorted phase of the seismic wave. In particular, when the hydrate saturation is low or the underlying formation does not contain free gas, it is difficult to observe the identifiable BSR in the traditional acoustic imaging profile, which causes difficulties in the identification of NGH. Here, we introduce the fractional viscoacoustic wave equation to perform the Q-compensated reverse-time migration (Q-RTM) for NGH, which can accurately recover the amplitude loss, correct the phase distortion, and provide high-resolution and high-illumination imaging results. Finally, Q-RTM can effectively enhance the BSR, reduce the uncertainty of hydrate identification, help to confirm the location and spatial distribution of the gas hydrate-bearing sediments, further refine the geological properties, and provide some theoretical basis for the exploitation and drilling of hydrate.
Seismic attenuation is usually described by a constant-Q (CQ) model that assumes the seismic quality factor (Q) is independent of the frequency. To simulate the attenuation behaviors of seismic waves, we develop a pseudo-spectral time-domain (PSTD) method to solve a CQ viscoacoustic wave equation. This method is nearly fourth-order accurate in time. Compared to the conventional temporal second-order PSTD method, the new PSTD scheme is verified to be more efficient. In some applications such as Q-compensated reverse-time migration (Q-RTM) and time-reversal imaging, one requires to simulate an anti-attenuation process. To realize this purpose, we switch our viscoacoustic PSTD modeling scheme into an amplitude-compensated PSTD modeling scheme by flipping the signs of the operators that dominate the amplitude loss. To control the numerical instability caused by high-frequency overcompensation in the amplitude-boosted modeling, we integrate a time-variant filter to the PSTD modeling scheme to suppress the high-frequency noise. Wavefield simulation examples in homogeneous media verify the temporal accuracy of our nearly fourth-order PSTD modeling scheme. A Q-RTM test of synthetic data is also presented to demonstrate the robustness of our amplitude-compensated PSTD modeling scheme.
Purpose
The finite element method (FEM) is used to calculate the two-dimensional anti-plane dynamic response of structure embedded in D’Alembert viscoelastic multilayered soil on the rigid bedrock. This paper aims to research a time-domain absorbing boundary condition (ABC), which should be imposed on the truncation boundary of the finite domain to represent the dynamic interaction between the truncated infinite domain and the finite domain.
Design/methodology/approach
A high-order ABC for scalar wave propagation in the D’Alembert viscoelastic multilayered media is proposed. A new operator separation method and the mode reduction are adopted to construct the time-domain ABC.
Findings
The derivation of the ABC is accurate for the single layer but less accurate for the multilayer. To achieve high accuracy, therefore, the distance from the truncation boundary to the region of interest can be zero for the single layer but need to be about 0.5 times of the total layer height of the infinite domain for the multilayer. Both single-layered and multilayered numerical examples verify that the accuracy of the ABC is almost the same for both cases of only using the modal number excited by dynamic load and using the full modal number of infinite domain. Using the ABC with reduced modes can not only reduce the computation cost but also be more friendly to the stability. Numerical examples demonstrate the superior properties of the proposed ABC with stability, high accuracy and remarkable coupling with the FEM.
Originality/value
A high-order time-domain ABC for scalar wave propagation in the D’Alembert viscoelastic multilayered media is proposed. The proposed ABC is suitable for both linear elastic and D’Alembert viscoelastic media, and it can be coupled seamlessly with the FEM. A new operator separation method combining mode reduction is presented with better stability than the existing methods.
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.
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.
Zoeppritz equations form the theoretical basis of most existing amplitude variation with incident angle (AVA) inversion methods. Assuming that only primary reflections exist, that is, the multiples are fully suppressed and the transmission loss and geometric spreading are completely compensated for, Zoeppritz equations can be used to solve for the elastic parameters of strata effectively. However, for thin interbeds, conventional seismic data processing technologies cannot suppress the internal multiples effectively, nor can they compensate for the transmission loss accurately. Therefore, AVA inversion methods based on Zoeppritz equations or their approximations are not applicable to thin interbeds. In this study, we propose a prestack AVA inversion method based on a fast algorithm for reflectivity. The fast reflectivity method can compute the full-wave responses, including the reflection, transmission, mode conversion, and internal multiples, which is beneficial to the seismic inversion of thin interbeds. A further advantage of the fast reflectivity method is that the partial derivatives of the reflection coefficient with respect to the elastic parameters can be expressed as analytical solutions. Based on the Gauss–Newton method, we construct the objective function and model-updating formula considering sparse constraint, where the Jacobian matrix takes the form of an analytical solution, which can significantly accelerate the inversion convergence. We validate our inversion method using numerical examples and field seismic data. The inversion results demonstrate that the fast reflectivity-based inversion method is more effective for thin interbed models in which the wave-propagation effects, such as interval multiples, are difficult to eliminate.
Theoretical analyses of propagation operators play an important role in the development of true-amplitude migration methods. We analyze the influence of the propagation effects, including geometric spreading and transmission losses, on the amplitudes of reverse time migration (RTM) in variable density and velocity media. Amplitude behaviors of the forward- and backward-propagation operators in RTM are studied with stationary phase theory. We prove that under the high-frequency approximation, geometric spreading in the forward propagation will be compensated automatically in backward propagation. We also prove that because of transmission losses, RTM formula contains an angle-dependent transmission coefficient item that prevents RTM from yielding a true-amplitude image of reflection coefficients. These derivations and proofs provide theoretical and mathematical base for the methods to compensate transmission losses with plane-wave transmission coefficients for true-amplitude RTM. Several numerical examples confirm that the compensation of transmission loss helps to improve the amplitude accuracy of true-amplitude RTM.
A 2D algorithm for angle-domain common-image gather calculation is extended and modified to produce 3D elastic angle and azimuth common-image gathers. The elastic seismic data are propagated with the elastic particle displacement wave equation, and then the PP-reflected and PS-converted waves are separated by divergence and curl calculations during application of the image condition, and the excitation-time image condition is applied. The incident angles and azimuths are calculated using source propagation directions and the reflector normals.The source propagation direction vector is computed as the spatial gradient of the incident three-component P-wavefield. The vector normal to the reflector is calculated using the Hilbert transform. Ordering the migrated images with respect to incident angles for a fixed azimuth bin, and with respect to azimuths for a fixed incident angle bin, creates angle-domain and azimuth-domain common-image gathers, respectively. Sorting the azimuth gathers by the incident angle bins causes a shift to a greater depth for too-high migration velocity and to a lesser depth for too-low migration velocity. In an azimuth angle gather, the inner loop is over azimuth and the outer loop is over incident angle. For the sorted incident angle gathers, the velocity-dependent depth moveout is within the angle gathers, and across azimuth gathers.This method is compared with three other 3D CIG algorithms with respect to the number of calculations, and their disk storage and RAM requirements; it is three to six orders of magnitude faster and requires two to three orders of magnitude less disk space. The method is successfully tested with data for a modified part of the SEG/EAGE overthrust model.
Comparing to primary arrivals, multiples have longer propagation paths and smaller reflection angles, leading to wider illumination area in the horizontal direction and higher resolution in the vertical direction. Hence, it will be a better choice to make full use of the multiples rather than suppressing them. However, seismic attenuation exists widely in the subsurface medium, especially directly below the deep sea bottom. Therefore, to compensate for the attenuation effect during the multiples imaging, we propose a viscoacoustic reverse time migration method of different-order multiples. Following the multiple propagation paths, we compensate for the attenuation during both source wavefield forward propagation and receiver backward propagation, and introduce a regularization operator to automatically eliminate the exponential high-frequency noise during the attenuation compensation process. Taking advantage of the full wavefield information, we jointly utilize the different-order multiples and primaries when implementing the viscoacoustic reverse time migration. In numerical examples, we validate the viscoacoustic reverse time migration of different-order multiples in a three-layer attenuation model and an attenuating Sigsbee2B model. Results suggest that the proposed method can image the models using different-order multiples separately, which suppresses the crosstalk artefacts, balances the energy, raises the resolution, and improves subsalt images dramatically.
Absorption decreases the amplitude of waves propagating in the earth. In addition, it narrows the bandwidth and modifies the phase. Obtaining true-amplitude migration and acceptable resolution in dissipative media requires amigration algorithm that corrects these effects. To achieve this, a reverse-time prestack depth migration technique is used. This uses a stable amplifying finite-difference wavefield extrapolation that compensates for the attenuating propagation of the wavefields in the real earth. Numerical tests on synthetic VSP data illustrate how dissipation destroys the amplitude, phase and resolution in the migration images, and how our algorithm corrects these effects. The tests also show that the method is stable with respect to noise, and does not require a very detailed macro-model for dissipation.
Prestack depth migration of shot profiles by down-ward continuation is a practical imaging algorithm that is especially cost-effective for sparse-shot wide-azimuth geometries. The interpretation of offset as the dis-placement between the downward-propagating (shot) wavefield and upward-propagating (receiver) wavefield enables us to extract offset-domain common image-point (CIP) gathers during shot-profile migration. The offset-domain gathers can then be transformed to the angle domain with a radial-trace mapping originally introduced for shot-geophone migration. The compu-tational implications of this procedure include both the additional cost of multioffset imaging and an im-plicit transformation from shot-geophone to midpoint-offset coordinates. Although this algorithm provides a mechanism for imaging angle-dependent reflectivity via shot-profile migration, for sparse-shot geometries the fundamental problem of shot-aliasing may severely im-pact the quality of CIP gathers.
In an anisotropic medium, a normal-incidence wave is multiply transmitted and reflected down to a reflector where the phase-velocity vector is parallel to the interface normal. The ray code of the upgoing wave is equal to the ray code of the downgoing wave in reverse order. The geometric spreading, KMAH index, and transmission and reflection coefficients of the normal-incidence ray can be expressed in terms of products or sums of the corresponding quantities of the one-way normal and normal-incidence-point (NIP) waves. Here, we show that the amplitude of the ray-theoretic Green's function for the reflected wave also follows a similar decomposition in terms of the amplitude of the Green's function of the NIP wave and the normal wave. We use this property to propose three schemes for true-amplitude poststack depth migration in anisotropic media where the image represents an estimate of the zero-offset reflection coefficient. The first is a map migration procedure in which selected primary zero-offset reflections are converted into depth with attached true amplitudes. The second is a ray-based, Kirchhoff-type full migration. The third is a wave equation continuation algorithm to reverse-propagate the recorded wavefield in a half-velocity model with half the elastic constants and double the density. The image is formed by taking the reverse-propagated wavefield at time equal to zero followed by a geometric spreading correction.
In prestack depth migration using explicit extrapolators, the attenuation and dispersion of the seismic wave has been neglected so far. The authors present a method for accommodating absorption and dispersion effects in depth migration schemes. Extrapolation operators that compensate for absorption and dispersion are designed using an optimization algorithm. The design criterion is that the wavenumber response of the operator should equal the true extrapolator. Both phase velocity and absorption macro models are used in the wavefield extrapolation. In a model with medium to high absorption, the images obtained are superior to those obtained using extrapolators without compensation for absorption.
We present two methods to compute angle-domain common image gathers (ADCIG) by downward-continuation migration, and we an-alyze their amplitude response versus reflection angle (AVA). A straightforward implementation of the two methods leads to contra-dictory, and thus obviously inaccurate, amplitude responses. The amplitude problem is related to the fact that downward continua-tion migration is the adjoint of upward-continuation modeling, but it is a poor approximation of its inverse. We derive the weight-ing operators, diagonal in the frequency-wavenumber domain, that makes migration a good approximation to the inverse of modeling. After weighting, the ADCIGs computed by the two methods be-come consistent. Other important factors degrading the accuracy of AVA in practical situation are the limited sampling and offset range, and the bandlimited nature of seismic data.
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.
True amplitude seismic data are desired for seismic interpretation because they contain direct hydrocarbon indicators. Amplitudes of reflections from deeper target zones are often differentially weakened and distorted by the spatially variable transmission and intrinsic attenuation (Q) losses in the overlying layers. A compensation function for transmission loss is directly estimated from the reflections recorded in surface seismic traces. To obtain the compensation function for attenuation loss, Q needs to be estimated a priori. This method is first tested on synthetic zero-offset data, then applied to a real 3-D stacked volume in which the target zone is masked by a group of shallow bright spots. This is a very efficient, physically-based algorithm that gives visually balanced amplitudes that are close to true relative amplitudes along the deeper target reflections.
True-amplitude Kirchhoff migration (TAKM) is an important tool in seismic-reflection imaging. In addition to a structural image, it leads to reflectivity maps of the subsurface. TAKM is carried out in terms of a weighted diffraction stack where the weight functions are computed with dynamic ray tracing (DRT) in addition to the diffraction traveltimes. DRT, however, is time-consuming and imposes restrictions on the velocity models, which are not always acceptable. An alternative approach to TAKM is proposed in which the weight functions are directly determined from the diffraction traveltimes. Because other methods exist for the generation of traveltimes, this approach is not limited by the requirements for DRT. Applications to a complex synthetic model and real data demonstrate that the image quality and accuracy of the reconstructed amplitudes are equivalent to those obtained from TAKM with DRT-generated weight functions.
Schemes for seismic mapping of reflectors in the presence of an arbitrary velocity model, dipping and curved reflectors, diffractions, ghosts, surface elevation variations, and multiple reflections are reviewed and reduced to a single formula involving up and downgoing waves. The mapping formula may be implemented without undue complexity by means of difference approximations to the relativistic Schroedinger equation.
One-way wave operators are powerful tools for forward modeling and migration. Here, we describe a recently developed true-amplitude implementation of modified one-way operators and present some numerical examples. By "true-amplitude" one-way forward modeling we mean that the solutions are dynamically correct as well as kinematically correct. That is, ray theory applied to these equations yields the upward- and downward-traveling eikonal equations of the full wave equation, and the amplitude satisfies the transport equation of the full wave equation. The solutions of these equations are used in the standard wave-equation migration imaging condition. The boundary data for the downgoing wave is also modified from the one used in the classic theory because the latter data is not consistent with a point source for the full wave equation. When the full wave-form solutions are replaced by their ray-theoretic approximations, the imaging formula reduces to the common-shot Kirchhoff inversion formula. In this sense, the migration is true amplitude as well. On the other hand, this new method retains all of the fidelity features of wave equation migration. Computer output using numerically generated data confirms the accuracy of this inversion method. However, there are practical limitations. The observed data must be a solution of the wave equation. Therefore, the data must be collected from a single common-shot experiment. Multi-experiment data, such as common-offset data, cannot be used with this method as presently formulated.
Wave propagation effects can significantly affect amplitude variation with offset (AVO) measurements. These effects include spreading losses, transmission losses, interbed multiples, surface multiple reflections, P-SV mode converted waves and inelastic attenuation. Examination of prestack elastic synthetic seismograms suggests that spreading losses and the transmission losses plus compressional interbed multiples are manifest mainly as a time and offset effect on the primary reflections. The surface related multiples and the P-SV mode-converted waves interfere with prestack amplitudes inducing distortions in the AVO pattern. Such distortions cause large variances in AVO model fitting. Prestack viscoelastic synthetic seismograms also suggest that inelastic attenuation further complicates the AVO response because of the offset and time variant amplitude decay effects and the phase change due to dispersion. Together, all these effects severely alter AVO behavior and result in serious errors in AVO parameter estimates being made from inadequately corrected seismograms. This modeling study suggests that time and offset dependent data processing prior to AVO analysis would be necessary to correct for the wave propagation effects, via either inverse filtering or model based approaches. Comparisons between acoustic and elastic synthetic seismograms show that corrections for the wave propagation effects derived using acoustic approximations are inadequate. Corrections need to be calculated based on elastic approximations provided that the inelastic attenuation effects have been previously removed.
The compressional wave reflection coefficient R( theta ) given by the Zoeppritz equations is simplified. The result is arranged into three terms which contribute to three distinct features of the R( theta ) curve: (1) the normal-incidence magnitude, (2) the behavior at intermediate angles of about 30 degrees, and (3) the approach to critical angle. Thus the author approximately diagonalize the multivariate relationship between elastic properties and curve features. The coefficient for intermediate angles has two terms: one term is proportional to TRIAN OP sigma , the contrast in Poisson's ratio; and the other term is TRIAN OP A//0, which describes the bland decrease of R( theta ) in the absence of contract in Poisson's ratio. When angles approaching critical are not included, R( theta ) may be adequately approximated by a parabola.
Since the first 3-D survey 17 years ago, companies of the Shell Group have operated over 250 such surveys outside North America and, as non-operating partners, have been involved in many others. The method's power to produce high-resolution images of the subsurface has been thoroughly proven, and it has become increasingly apparent that subsurface models resulting from mapping with 2-D seismic lack the detail provided by the 3-D method. Also, amplitude effects caused by variations in porefill and/or lithology and more easily seen on 3-D seismic than on 2-D profiles. Thus, in addition to its initial use for structural mapping, 3-D seismic is used increasingly to evaluate trends in reservoir quality.
The trends and variance in amplitude variation with offset (AVO) observations caused by near-surface structure, attenuation, and scattering are numerically synthesized by pseudospectral viscoelastic 2-D modeling. Near-surface structure produces amplitude focusing and defocusing that significantly distort AVO observations in offset windows at a scale comparable to that of the lateral variations in the structure. Attenuation and scattering decrease absolute amplitudes at all offsets. Scattering and wave interference increase the variance associated with AVO measurements. Depending on the relative influence of intrinsic attenuation, apparent attenuation associated with scattering, and geometrical focusing, a normalized AVO response can increase or decrease with offset (relative to that for the associated elastic, nonscattering, 1-D solution), and so mimic the behavior predicted as a function of contrasts in density, velocity, porosity or Poisson's ratio. If only relative (normalized) amplitudes are available, it is difficult to distinguish between effects of parameters whose main contributions are to absolute amplitude; for example, a trend of decreasing amplitude (relative to that for an elastic flat-layered model) produced by intrinsic attenuation may be counteracted by focusing/scattering or anisotropic effects over wide aperture ranges. Diagnostic information on AVO effects of scattering and attenuation is lost when the noise level is sufficiently high. Interpretations of AVO observations based on homogeneous layered elastic models must therefore be used with caution as they are, in general, nonunique. Lateral variations in AVO parameters are the key to detecting hydrocarbons, so lateral changes in AVO produced by lateral changes in the overburden properties have potential for being misinterpreted, especially if the recording aperture is small.
Kirchhoff migration has traditionally been viewed as an imaging procedure. Usually, few claims are made regarding the amplitudes in the imaged section. In recent years, a number of inversion formulas, similar in form to those of Kirchhoff migration, have been proposed. A Kirchhoff‐type inversion produces not only an image but also an estimate of velocity variations, or perhaps reflection coefficients. The estimate is obtained from the peak amplitudes in the image. In this paper prestack Kirchhoff migration and inversion formulas for the one‐parameter acoustic wave equation are compared. Following a heuristic approach based on the imaging principle, a migration formula is derived which turns out to be identical to one proposed by Bleistein for inversion. Prestack Kirchhoff migration and inversion are, thus, seen to be the same—both in terms of the image produced and the peak amplitudes of the output.
The classical interpretation relating Amplitude Versus Offset (AVO) to Poisson's ratio and other petrophysical properties is based on the assumptions of elasticity and isotropy. We extend this interpretation to a layered medium with anisotropic and/or viscoelastic properties, using a Fourier Pseudo-Spectral method to solve the wave equation. Both viscoelasticity and anisotropy are key factors for the quantitative interpretation of AVO trends, because they contribute to the seismic energy partition at geological interfaces (the reflection coefficients), and because they continually induce propagation effects. We show that: 1) Reflection coefficients at an interface are strongly dependent on the elastic anisotropy of both the overlying and the underlying media. The AVO effect is further complicated by materials with viscoelastic properties. 2) Propagation effects are due to elastic anisotropic energy focusing and viscoelastic dissipation that distort the energy and phase distribution of the incident and reflected wavefronts. These two phenomena can be of the same order of magnitude as variations in reflection amplitudes with offset and can make it difficult to recover reflection coefficients along an interface from seismic data. In theory, they make the amplitude determination of a seismic event somewhat dependent on wavelet phase changes that occur continually as the wavefront propagates. In practice, they create anisotropic radiation patterns and differentially focus the seismic energy distribution along the wavefront. For these reasons, all detailed reservoir characterizations based on modeling and interpretation work should attempt to account for anisotropy and viscoelastic attenuation; this is not an easy task in the real world because of the difficulty in prescribing appropriate physical parameters.
To apply reverse-time migration to prestack, finite-offset data from variable-velocity media, the standard (time zero) imaging condition must be generalized because each point in the image space has a different image time (or times). This generalization is the excitation-time imaging condition, in which each point is imaged at the one-way traveltime from the source to that point.Reverse-time migration with the excitation-time imaging condition consists of three elements: (1) computation of the imaging condition; (2) extrapolation of the recorder wave field; and (3) application of the imaging condition. Computation of the imaging condition for each point in the image is done by ray tracing from the source point; this is equivalent to extrapolation of the source wave field through the medium. Extrapolation of the recorded wave field is done by an acoustic finite-difference algorithm. Imaging is performed at each step of the finite-difference extrapolation by extracting, from the propagating wave field, the amplitude at each mesh point that is imaged at that time and adding these into the image space at the same spatial locations. The locus of all points imaged at one time step is a wavefront [a constant time (or phase) trajectory]. This prestack migration algorithm is very general. The excitation-time imaging condition is applicable to all source-receiver geometries and variable-velocity media and reduces exactly to the usual time-zero imaging condition when used with zero-offset surface data. The algorithm is illustrated by application to both synthetic and real VSP data. The most interesting and potentially useful result in the processing of the synthetic data is imaging of the horizontal fluid interfaces within a reservoir even when the surrounding reservoir boundaries are not well imaged.
With the increasing ambition of characterizing hydrocarbon traps in more subtle or complex reservoirs, Amplitude Variation with Offset (AVO) techniques are becoming a valuable seismic tool for quantitative seismic discrimination of lithologies and fluids. One of the biggest remaining challenges is to acquire and process the data in an amplitude preserved fashion and in multi-dimensional geology. This study is a component of this puzzle, and attempts to address the following processing question: what are the benefits of prestack migration before AVO inversion (process 1) versus performing an AVO inversion followed by a poststack migration (process 2)? The comparison is done on a 2-D synthetic model which is valid for process 2. The technique used for process 1 is the prestack depth AVO migration/inversion described in the text which estimates reflectivities and incidence angles in multi-dimensions from the data prior to AVO inversion. Process 2 results are derived using a commercial seismic processing software package.
Most present day seismic migration schemes determine only the zero-offset reflection coefficient for each grid point (depth point) in the subsurface. Our objective is to obtain angle-dependent reflection coefficients from seismic data by means of prestack migration (multisource, multioffset). After downward extrapolation of source and reflected wave fields to one depth level, the rows of the reflectivity matrix (representing angle-dependent reflectivity information for each grid point at that depth level) are recovered by deconvolving the reflected wave fields with the related source wave fields. This process is carried out in the space-frequency domain. The new imaging technique has been tested on media with horizontal layers. However, with our shot-record orientated algorithm it is possible to handle any subsurface geometry. The first tests show excellent results up to high angles, both in the acoustic and in the elastic case. With angle-dependent reflectivity information it becomes feasible to derive detailed velocity and density information in a subsequent stratigraphic inversion step. -from Authors
This paper treats the linearized inverse scattering problem for the case of variable background velocity and for an arbitrary configuration of sources and receivers. The linearized inverse scattering problem is formulated in terms of an integral equation in a form which covers wave propagation in fluids with constant and variable densities and in elastic solids. This integral equation is connected with the causal generalized Radon transform (GRT), and an asymptotic expansion of the solution of the integral equation is obtained using an inversion procedure for the GRT. The first term of this asymptotic expansion is interpreted as a migration algorithm. As a result, this paper contains a rigorous derivation of migration as a technique for imaging discontinuities of parameters describing a medium. Also, a partial reconstruction operator is explicitly derived for a limited aperture. When specialized to a constant background velocity and specific source–receiver geometries our results are directly related to some known migration algorithms.
Knowledge of elastic parameter (compressional and shear velocities and density) contrasts within the earth can yield knowledge of lithology changes. Elastic parameter contrasts manifest themselves on seismic records as angle-dependent reflectivity. Interpretation of angle-dependent reflectivity, or amplitude variation with offset (AVO), on unmigrated records is often hindered by the effects of common-depth-point smear, incorrectly specified geometrical spreading loss, source/receiver directivity, as well as other factors. It is possible to correct some of these problems by analyzing common-reflection-point gathers after prestack migration, provided that the migration is capable of undoing all the amplitude distortions of wave propagation between the sources and the receivers. A migration method capable of undoing such distortions and thus producing angle-dependent reflection coefficients at analysis points in a lossless, isotropic, elastic earth is called a "true-amplitude migration." The principles of true-amplitude migration are simple enough to allow several methods to be considered as "true-amplitude." I consider three such migration methods in this paper: one associated with Berkhout, Wapenaar, and co-workers at Delft University; one associated with Bleistein, Cohen, and co-workers at Colorado School of Mines and, more recently, Hubral and co-workers at Karlsruhe University; and a third introduced by Tarantola and developed internationally by many workers. These methods differ significantly in their derivations, as well as their implementation and applicability. However, they share some fundamental similarities, including some fundamental limitations. I present and compare summaries of the three methods from a unified perspective. The objective of this comparison is to point out the similarities of these methods, as well as their relative strengths and weaknesses.
True-amplitude wave-equation migration provides a quality migrated image of the earth's interior. In addition, the amplitude of the output provides an estimate of the angular-dependent reflection coefficient, similar to the output of Kirchhoff inversion. Recently, true-amplitude wave-equation migration for common-shot data has been proposed to generate amplitude-reliable, shot-domain, common-image gathers in heterogeneous media. We present a method to directly produce angle-domain common-image gathers from both common-shot and shot-receiver wave-equation migration. Generating true-amplitude, shot-domain, common-image gathers requires a deconvolution-type imaging condition using the ratio of the upgoing and downgoing wavefield, each downward-projected to the image point. Producing true-amplitude, angle-domain, common-image gathers requires, instead, the product of the upgoing wavefield and the complexconjugate of the downgoing wavefield in the imaging condition. Since multiplication is a more stable computational process than division, the new methods proposed provide more stable ways of inverting seismic data. Furthermore, the resulting common-image gathers can be directly used for migrated amplitude-variation-with angle analysis and tomography-based velocity analysis. Shot-receiver wave-equation migration requires new true-amplitude, one-way wave equations with one depth variable and transverse variables for the coordinates corresponding to sources and receivers, hence, two transverse coordinates in 2D and four transverse coordinates in 3D. We propose a modified double-square-root one-way wave equation to produce true amplitude common-image angle gathers. We also demonstrate the new methods with some synthetic examples. Some numerical examples show that the new methods we propose give better amplitude performance on the migrated angle gathers.
Unmigrated reflection seismic energy recorded at the earth’s surface, for example in the form of common‐offset records, often provides a remarkably coherent picture of subsurface structure. Coherent as it might be, however, this picture is not correct, suffering from several distorting effects, most notably those of diffraction from geologic bed truncations and lateral movement of the energy between the reflection points on dipping beds and the surface locations. Seismic imaging is the process that corrects these distortions. In the next paragraphs, I discuss this important technology.
Three different theoretical approaches to amplitude-preserving Kirchhoff depth migration are compared. Each of them suggests applying weights in the diffraction stack migration to correct for amplitude loss resulting from geometric spreading. The weight functions are given in different notations, but as is shown, all of these expressions are similar. A notation that is well suited for implementation is suggested: entirely in terms of Green's function quantities (amplitudes or point-source propagators). For the most common prestack configurations (common-shot and common-offset) and 3-D, 2.5-D, and 2-D migrations, expressions of the weights are given in this notation. The quantities needed for calculation of the weights can be computed easily, e.g., by dynamic ray tracing.
In this paper, I present a modification of the Beylkin inversion operator. This modification accounts for the band-limited nature of the data and makes the role of discontinuities in the sound speed more precise. The inversion presented here partially dispenses with the small-parameter constraint of the Born approximation. This is shown by applying the proposed inversion oper- ator to upward scattered data represented by the Kirch- hoff approximation, using the angularly dependent geometrical-optics reflection coefficient. A fully nonlin- ear estimate of the jump in sound speed may be extrac- ted from the output of this algorithm interpreted in the context of these Kirchhoff-approximate data for the for- ward problem. The inversion of these data involves integration over the source-receiver surface, the reflecting surface, and frequency. The spatial integmis are computzd by Thor method of stationary phase. The output is asymp-
Kirchhoff prestack depth migration is an accepted approach for converting surface seismic data into an image of the subsurface. One problem with this approach is that it produces arbitrary images. The true-amplitude theory, and the infinite-frequency ray-traced amplitudes overcome this problem. However, performing both methods is costly. For this application, the complete true-amplitude theory can be fully implemented without having to give up the efficiency of standard amplitude-oblivious processing.
The authors study the influence of elastic 1-D inhomogeneous random media (e.g., finely layered media with variable density and shear and compressional velocities) on the kinematics and dynamics of the transmitted obliquely incident P- and SV-plane waves. Multiple scattering (resulting in localization and spatial dispersion of the elastic wavefield) is the main physical effect controlling the properties of the wavefield in such media. The authors analyze the wave propagation assuming the fluctuations of velocities and density to be small (of the order of 20% or smaller). They obtain explicit analytic solutions for the attenuation coefficient and phase velocity of the transmitted waves. These solutions are valid for all frequencies. They agree very well with results of numerical modeling. Their theory shows that fine elastic multilayering is characterized by a frequency-dependent anisotropy. At typical acquisition frequencies this anisotropy differs significantly from the low-frequency anisotropy described by the well-known Backus averaging. The increase of the phase velocity with frequency is quantified. It can partly explain the difference between well-log-derived velocities and lower frequency seismic velocities in terms of localization. The low- and high-frequency asymptotical results for the phase velocity agree with those of Backus averaging and ray approximation, respectively. The theory describes the angle-dependent attenuation caused by multiple scattering. The proposed formulas are simple enough to be used in many practical applications as, e.g., in an amplitude variation with offset (AVO) analysis. They can be implemented for taking into account the angle dependence of transmission effects, or they can be used in an inversion for statistical parameters of sediments.
One-way wave operators are powerful tools for use in forward modelling and inversion. Their implementation, however, involves introduction of the square root of an operator as a pseudo-differential operator. Furthermore, a simple factoring of the wave operator produces one-way wave equations that yield the same travel times as the full wave equation, but do not yield accurate amplitudes except for homogeneous media and for almost all points in heterogeneous media. Here, we present augmented one-way wave equations. We show that these equations yield solutions for which the leading order asymptotic amplitude as well as the travel time satisfy the same differential equations as the corresponding functions for the full wave equation. Exact representations of the square-root operator appearing in these differential equations are elusive, except in cases in which the heterogeneity of the medium is independent of the transverse spatial variables. Here, we address the fully heterogeneous case. Singling out depth as the preferred direction of propagation, we introduce a representation of the square-root operator as an integral in which a rational function of the transverse Laplacian appears in the integrand. This allows us to carry out explicit asymptotic analysis of the resulting one-way wave equations. To do this, we introduce an auxiliary function that satisfies a lower dimensional wave equation in transverse spatial variables only. We prove that ray theory for these one-way wave equations leads to one-way eikonal equations and the correct leading order transport equation for the full wave equation. We then introduce appropriate boundary conditions at z = 0 to generate waves at depth whose quotient leads to a reflector map and an estimate of the ray theoretical reflection coefficient on the reflector. Thus, these true amplitude one-way wave equations lead to a 'true amplitude wave equation migration' (WEM) method. In fact, we prove that applying the WEM imaging condition to these newly defined wavefields in heterogeneous media leads to the Kirchhoff inversion formula for common-shot data when the one-way wavefields are replaced by their ray theoretic approximations. This extension enhances the original WEM method. The objective of that technique was a reflector map, only. The underlying theory did not address amplitude issues. Computer output obtained using numerically generated data confirms the accuracy of this inversion method. However, there are practical limitations. The observed data must be a solution of the wave equation. Therefore, the data over the entire survey area must be collected from a single common-shot experiment. Multi-experiment data, such as common-offset data, cannot be used with this method as currently formulated. Research on extending the method is ongoing at this time.
True-amplitude (TA) migration, which is a Kirchhoff-type modified weighted diffraction stack, recovers (possibly) complex angle-dependent reflection coefficients which are important for amplitude-versus-offset (AVO) inversion. The method can be implemented using existing prestack or post-stack Kirchhoff migration and fast Green's function computation programs. Here, it is applied to synthetic single-shot and constant-offset seismic data that include post-critical reflections (complex reflection coefficients) and caustics. Comparisons of the amplitudes of the TA migration image with theoretical reflection coefficients show that the (possibly complex) angle-dependent reflection coefficients are correctly estimated.
Attenuation measurements were made near Limon, Colorado, where the Pierre shale is unusually uniform from depths of less than 100 ft to approximately 4,000 ft. Particle velocity wave forms were measured at distances up to 750 ft from explosive and mechanical sources. Explosives gave a well‐defined compressional pulse which was observed along vertical and horizontal travel paths. A weight dropped on the bottom of a borehole gave a horizontally‐traveling shear wave with vertical particle motion. In each case, signals from three‐component clusters of geophones rigidly clamped in boreholes were amplified by a calibrated, wide‐band system and recorded oscillographically. The frequency content of each wave form was obtained by Fourier analysis, and attenuation as a function of frequency was computed from these spectra. For vertically‐traveling compressional waves, an average of 6 determinations over the frequency range of 50–450 cps gives α=0.12 f. For horizontally‐traveling shear waves with vertical motion in the frequency range 20–125 cps, the results are expressed by α=1.0 f. In each case attenuation is expressed in decibels per 1,000 ft of travel and f is frequency in cps. These measurements indicate, therefore, that the Pierre shale does not behave as a visco‐elastic material.
A bstract
The amplitude of seismic energy varies over a tremendous range. Some of the factors responsible for such variation do not contain subsurface information; these include source strength and coupling, geophone sensitivity, array directivity, instrument balance, scattering in the near‐surface, for example. Others depend on subsurface factors but do not convey information about lithology or hydrocarbon accumulation in a form from which we are able to extract it; these include spherical divergence, ray‐path curvature, loss in transmission through intervening reflectors, peg‐leg multiples, reflector rugosity, and curvature. The amplitude‐governing factors we are primarily interested in are reflection coefficient, the interference of reflections from the top and base of a sand, and absorption.
The nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation. The method is based on the generalized least-squares criterion. The inverse problem can be solved using an iterative algorithm which gives, at each iteration, updated values of bulk modulus, density, and time source function. Each step of the iterative algorithm essentially consists of a forward propagation of the actual sources in the current model and a forward propagation (backward in time) of the data residuals. The correlation at each point of the space of the two fields thus obtained yields the corrections of the bulk modulus and density models. This shows, in particular, that the general solution of the inverse problem can be attained by methods strongly related to the methods of migration of unstacked data and commercially competitive with them. Refs.
To account for elastic and attenuating effects in the elastic wave equation, the stress-strain relationship can be defined through a general, anisotropic, causal relaxation function
ijkl
(x, ). Then, the wave equation operator is not necessarily symmetric (self-adjoint), but the reciprocity property is still satisfied. The representation theorem contains a term proportional to the history of strain. The dual problem consists of solving the wave equation withfinal time conditions and an anti-causal relaxation function. The problem of interpretation of seismic waveforms can be set as the nonlinear inverse problem of estimating the matter density (x) and all the functions
ijkl
(x, ). This inverse problem can be solved using iterative gradient methods, each iteration consisting of the propagation of the actual source in the current medium, with causal attenuation, the propagation of the residuals—acting as if they were sources—backwards in time, with anti-causal attenuation, and the correlation of the two wavefields thus obtained.
Oz Yilmaz has expanded his original volume on processing to include inversion and interpretation of seismic data. In addition to the developments in all aspects of conventional processing, this two-volume set represents a comprehensive and complete coverage of the modern trends in the seismic industry-from time to depth, from 3-D to 4-D, from 4-D to 4-C, and from isotropy to anisotropy.
Seismic impedance: First Break
- K Helbig
Helbig, K., 1983, Seismic impedance: First Break, 1, 25–32.
AVO migration and inversion: Are they commutable?: 64th Annual International Meeting, SEG, Expanded Abstracts
- W B Beydoun
- S Jin
- C Hanitzsch
Beydoun, W. B., S. Jin, and C. Hanitzsch, 1994, AVO migration and inversion: Are they commutable?: 64th Annual International Meeting, SEG, Expanded Abstracts, 952–955.
Ray method in seismology: Univerzita Karlova, 192. Červený, V., and R. Ravindra, 1971, Theory of seismic head waves
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Červený, V., I. A. Molotkov, and I. Pšenčik, 1977, Ray method in seismology: Univerzita Karlova, 192. Červený, V., and R. Ravindra, 1971, Theory of seismic head waves: Univer-sity of Toronto Press.