Q-compensated Reverse Time Migration Research Articles

Power-law frequency-dependent Q simulations in viscoacoustic media using decoupled fractional Laplacians

Quantifying the seismic attenuation of wave propagation in the earth’s interior is essential for studying subsurface structures. Previous approaches for attenuation simulations (e.g., the standard linear solid and the fractional derivative model) are mainly based on the frequency-independent quality factor Q assumption. However, seismic attenuation in high-temperature and high-pressure regions usually exhibits power-law frequency-dependent Q characteristics. To simulate this Q effect in attenuative media, we derive a new viscoacoustic wave equation with decoupled fractional Laplacians in the time domain. Unlike the existing methods using relaxation functions to fit the power-law relationship in a specific frequency band, our equation is directly derived from the approximated complex modulus, which explicitly involves the reference quality factor and fractional exponent parameters. Furthermore, this equation contains two fractional Laplacians, which can easily simulate decoupled amplitude dissipation and phase distortion effects, making it amenable to Q-compensated reverse time migration. In the implementation, a Taylor-series expansion and a pseudospectral method are introduced to solve the fractional Laplacians with variable fractional exponents. Numerical experiments demonstrate the effectiveness of our method for power-law frequency-dependent Q simulations. As a forward-modeling engine, our derived viscoacoustic wave equation is a good supplement to the current Q simulation methods and it could be applied in many seismic applications, such as Q-compensated reverse time migration and full-waveform inversion.

Truncated pseudo-differential operator √−▽2 and its applications in viscoacoustic reverse-time migration

SUMMARY The pseudo-differential operator with symbol |k|α has been widely used in seismic modelling and imaging when involving attenuation, anisotropy and one-way wave equation, which is usually calculated using the pseudo-spectral method. For large-scale problems, applying high-dimensional Fourier transforms to solve the wave equation that includes pseudo-differential operators is much more expensive than finite-difference approaches, and it is not suitable for parallel computing with domain decomposition. To mitigate this difficulty, we present a truncated space-domain convolution method to efficiently compute the pseudo-differential operator $\sqrt{-\nabla ^2}$, and then apply it to viscoacoustic reverse-time migration. Although $\sqrt{-\nabla ^2}$ is theoretically non-local in the space domain, we take the limited frequency band of seismic data into account, and constrain the approximated convolution stencil to a finite length. The convolution coefficients are computed by solving a least-squares inverse problem in the wavenumber domain. In addition, we exploit the symmetry of the resulting convolution stencil and develop a fast spatial convolution algorithm. The applications of the proposed method in Q-compensated reverse-time migration demonstrate that it is a good alternative to the pseudo-spectral method for computing the pseudo-differential operator $\sqrt{-\nabla ^2}$, with almost the same accuracy but much higher efficiency.

Power-Law Frequency-Dependent Q Simulation and Reverse-Time Migration in Transversely Isotropic Acoustic Media

ABSTRACT Velocity and attenuation (Q) anisotropy are widely distributed in the Earth’s interior, significantly affecting the kinematic and dynamic characteristics of seismic-wave propagations. Previous studies developed to simulate these effects are mainly restricted to the constant-Q assumption. However, seismic attenuation in high-temperature and high-pressure regions is demonstrated to be frequency-dependent and usually follows a power-law formulation. To simulate this Q effect in transversely isotropic (TI) attenuating media, we derive a new pure-viscoacoustic wave equation with decoupled fractional Laplacians, which can simultaneously simulate amplitude dissipation and velocity dispersion effects. Based on the wavenumber relationship between the observation and physical coordinate systems, the tilted TI (TTI) wave equation is further derived. Compared with the pseudoviscoacoustic wave equation, the proposed pure-viscoacoustic equation can simulate stable P wavefields in complex geological structures without S-wave artifacts. To solve this new equation, two low-rank decompositions are introduced to approximate the real and imaginary parts and avoid the separation of wavenumbers and dip angles, making it much simpler in programming and implementation. We further use this equation to perform Q-compensated reverse-time migration to generate high-resolution migration images in anisotropic attenuating media. Numerical examples demonstrate the effectiveness of the proposed method for pure-viscoacoustic wavefield simulations and migrations in TTI attenuating media with power-law frequency-dependent Q effects.

Modified pure-viscoacoustic wave propagation and compensated reverse-time migration in transversely isotropic media

Previous studies demonstrated that seismic attenuation and anisotropy can significantly affect the kinematic and dynamic characteristics of wavefields. If these effects are not incorporated into seismic migration, the resolution of the imaging results will be reduced. Considering the anisotropy of velocity and attenuation, we derive a new pure-viscoacoustic wave equation to simulate P wave propagation in transversely isotropic (TI) attenuating media by combining the complex dispersion relation and modified complex modulus. Compared to the conventional complex modulus, the modified modulus is derived from the optimized relationship between angular frequency and wavenumber, which can improve the modeling accuracy in strongly attenuating media. Wavefield comparisons illustrate that our pure-viscoacoustic wave equation can simulate stable P wavefields in complex geological structures without S-wave artifacts and generate similar P wave information to the pseudo-viscoacoustic wave equation. During the implementation, we introduce two low-rank decompositions to approximate the real and imaginary parts and then use the pseudo-spectral method to solve this new equation. Since the proposed equation can simulate decoupled amplitude attenuation and phase dispersion effects, it is used to perform Q-compensated reverse-time migration (Q-RTM). Numerical examples demonstrate the accuracy and robustness of the proposed method for pure-viscoacoustic wavefield simulations and migration imaging in transversely isotropic attenuating media.

A Multi-Task Learning Framework of Stable Q-Compensated Reverse Time Migration Based on Fractional Viscoacoustic Wave Equation

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.

A stable Q-compensated reverse-time migration method using a modified fractional viscoacoustic wave equation

Seismic wave propagation inside the earth is usually accompanied by amplitude dissipation and velocity dispersion. It is essential to accurately characterize these effects and make the corresponding compensations in seismic reverse-time migration (RTM) and the following interpretation. Due to the approximation of [Formula: see text] in the derivation, the conventional fractional viscoacoustic wave equation in strongly attenuative media is unsatisfactory. This paper develops a modified viscoacoustic wave equation by combining the relationship between angular frequency and complex wavenumber. The numerical simulation demonstrates its advantage of high accuracy in describing constant- Q effects, especially in strongly attenuative media with small Q values. A truncated Taylor-series expansion algorithm and a pseudospectral method are developed to solve this equation during wave propagation. To address the issue of numerical instability in Q-compensated RTM ( Q-RTM), a stabilized technology with regularization is further developed. Unlike the conventional filtering method implemented in the wavenumber domain, our approach only adds an explicit regularization term, which can be directly computed in the space domain. Furthermore, this regularization depends on velocities and quality factors, significantly improving its applicability in complicated structures. Noise-free data tests demonstrate that our Q-RTM method can effectively correct phase distortion, compensate for amplitude attenuation, and produce high-resolution migrated images without any cutoff-frequency artifacts caused by filtering. Noisy data experiments further verify the antinoise performance and robustness of our method.

Compensating attenuation effects in full-waveform inversion with dissipation-dispersion decoupling

Seismic attenuation poses challenges to velocity model building from acoustic/elastic full-waveform inversion (FWI). In particular, when constructing the FWI gradient for velocity inversion, recent studies have indicated that the high-attenuation structure could distort the gradient by damping the amplitudes and shifting the kinematic phases, resulting in an imbalanced update and thus unreliable velocity model. These Q effects are particularly significant in the reflection acquisition geometry due to the “double-damping” issue. Here, we develop a Q-compensated FWI algorithm for constructing a Q-free FWI gradient. By using a recently developed viscoacoustic wave propagator, this compensation can be done conveniently by keeping the dispersion term and flipping the dissipation term in the wave equation when we simulate the forward and time-reversed adjoint wavefields. The resultant gradient obtained by interacting these two wavefields has correct kinematics and Q-free amplitudes. This Q-compensated FWI can balance the update between Q- and no- Q-areas, which we determine using synthetic examples. In addition, we illustrate how to take advantage of the dissipation-dispersion decoupling to determine the anomalous Q value in the Q model building workflow via Q-compensated reverse time migration.

An explicit stable Q-compensated reverse time migration scheme for complex heterogeneous attenuation media

Prestack reverse-time migration (RTM) is a popular imaging technique for complex geological conditions, since the amplitude attenuation and velocity dispersion are common in seismic recordings. To image attenuated seismic recordings accurately, a robust migration algorithm with a stable attenuation compensation approach should be considered. In the context of the Q-compensated RTM approach based on the decoupled fractional Laplacians (DFLs) viscoacoustic wave equation, amplitude compensation can be implemented by flipping the sign of the dissipation term. However, the non-physical magnification of image amplitude could lead to a well-known numerical instability problem. The explicit stabilization operator can rectify the amplitude attenuation and suppress the numerical instability. However, limited by the inconvenient mixed-domain operator, the average Q value rather than the real Q value is often used in the compensation operator, lowering the compensated accuracy of the migration image. To overcome this problem, we propose a novel explicit Q-compensation scheme. The main advantage of the proposed compensation operator is that its order is space-invariant, making it more suitable for handling complex heterogeneous attenuation media. Several two-dimensional (2D) and three-dimensional (3D) synthetic models are used to verify the superiority of the proposed approach in terms of amplitude fidelity and image resolution. Field data further demonstrates that our approach has potential applications and can greatly enhance the resolution of seismic images.

Target-oriented Q-compensated reverse-time migration by using optimized pure-mode wave equation in anisotropic media

Research on seismic anisotropy and attenuation plays a significant role in exploration geophysics. To enhance the imaging quality for complicated structures, we develop several effective improvements for anisotropic attenuation effects in reverse-time migration (Q-RTM) on surface and vertical seismic profiling (VSP) acquisition geometries. First, to suppress pseudo-shear wave artifact and numerical instability of the commonly used anisotropic pseudo-acoustic wave equations, an optimized pure P-wave dispersion relation is derived and the corresponding pure-mode wave equation is solved by combining the finite-difference and Possion methods. Second, a simplified anisotropic pure-mode visco-acoustic wave equation (PVAWE) based on standard linear solid model is established. Third, a time-dispersion correlation strategy is applied to improve the modeling accuracy. Fourth, we extend a target-oriented scheme to anisotropic attenuated modeling and imaging. Instead of the conventional wavefield modeling and RTM, the proposed approach can extract available wavefield information near the target regions and produce high imaging resolution for target structures. Last, both anisotropic surface and VSP Q-RTMs are executed by combining optimized PVAWE, time-dispersion correlation and target-oriented algorithm. Modeling examples demonstrate the advantages of our schemes. Moreover, our modified Q-compensated imaging workflow can be regarded as a supplement to the classical anisotropic RTM.

Frequency-Domain Reverse-Time Migration With Attenuation Compensation

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 reverse time migration 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 anti-noise property of our method. The field data applications further prove its feasibility and practicability.

Frequency-Domain Q-Compensated Reverse Time Migration Using a Stabilization Scheme

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.

Frequency Domain Q-Compensated Reverse Time Migration Using Anti-Dispersion Acoustic Equation

Frequency Domain Q-Compensated Reverse Time Migration Using Anti-Dispersion Acoustic Equation

Accurately Stable Q-Compensated Reverse-Time Migration Scheme for Heterogeneous Viscoelastic Media

The development of multi-component seismic acquisition technology creates new possibilities for the high-precision imaging of complex media. Compared to the scalar acoustic wave equation, the elastic wave equation takes the information of P-waves, S-waves, and converted waves into account simultaneously, enabling accurate description of actual seismic propagation. However, inherent attenuation is one of the important factors that restricts multi-component high-precision migration imaging. Its influence is mainly reflected in the following three ways: first, the attenuation of the amplitude energy makes the deep structure display unclear; second, phase distortion introduces errors to the positioning of underground structures; and third, the loss of high frequency components reduces imaging resolution. Therefore, it is crucial to fully consider the absorption and attenuation characteristics of the real Earth during seismic modeling and imaging. This paper aims to develop an accurate attenuation compensation reverse-time migration scheme for complex heterogeneous viscoelastic media. We first utilize a novel viscoelastic wave equation with decoupled fractional Laplacians to depict the Earth’s attenuation behavior. Then, an adaptive stable attenuation compensation operator is developed to realize high-precision attenuation compensation imaging. Several synthetic and field data analyses verify the effectiveness of the proposed method.

Q-compensated viscoelastic reverse time migration in crosswell seismic imaging

Abstract Viscoelastic reverse time migration (RTM) is a practical approach to compensats for attenuation effects in migration, commonly studied in surface seismic research. Compared with conventional surface seismic imaging, the sources and receivers of crosswell seismic imaging are distributed in different wells with higher main frequencies and lower surface noises. The finer structures can be imaged. In this paper, we apply attenuation compensated RTM in crosswell migration to improve the resolution of images in attenuation regions. We solve the viscoelastic equations using the constant Q model because of its advantage of decoupling the amplitude and phase. The fractional-order amplitude and phase Laplacian operators are used. By spectral ration techniques, we observe the relationship between attenuation and frequency in crosswell seismic imaging. We describe the alignment and muting of direct P- and S-waves from full waves in detail. Synthetic examples demonstrate that Q-RTM images the geological structure more clearly than the traditional RTM. We reveal that compensated images describe the three-layer model, tilted fractures and randomly distributed caves more accurately. As the tilted angles increase, only two end points of the tilted fracture can be identified by conventional RTM, the boundary is blurred. We find that caves far from the source well can barely be imaged. By comparison, the compensated RTM approach obtains precise migration results of tilted fractures with arbitrary dip angles and caves. Finally, to approximate the real measurement data, noise is added to the viscoelastic data of the five-layer model. The effective noise immunity in crosswell seismic applications is verified.

Q-compensated reverse-time migration for natural gas hydrate using fractional viscoacoustic wave equation

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.

Constant-Q wave propagation and compensation by pseudo-spectral time-domain methods

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.

Q-compensated reverse time migration in tilted transversely isotropic media

Anisotropy and absorption are critical to the modeling and analysis of seismic amplitude,phase, and traveltime data. To neglect any of these phenomena, which are often bothoperating simultaneously, degrades the resolution and interpretability of migrated images.However, a full accounting of anisotropy and anelasticity is computationally complex andexpensive. One strategy for accommodating these aspects of wave propagation, while keepingcost and complexity under control, is to do so within an acoustic approximation. Weset up a procedure for solving the time-domain viscoacoustic wave equation for tilted transverselyisotropic (TTI) media, based on a standard linear solid model and, from this, developa viscoacoustic reverse time migration (Q-RTM) algorithm. In this approach, amplitudecompensation occurs within the migration process through a manipulation of attenuationand phase dispersion terms in the time domain differential equations. Specifically, theback-propagation operator is constructed by reversing the sign only of the amplitude lossoperators, but not the dispersion-related operators, a step made possible by reformulatingthe absorptive TTI equations such that the loss and dispersion operators appear separately.The scheme is tested on synthetic examples to examine the capacity of viscoacoustic RTM to correct for attenuation, and the overall stability of the procedure.

A Novel Wavefield-Reconstruction Algorithm for RTM in Attenuating Media

Q-compensated reverse-time migration ( Q-RTM) has been proven as an efficient method for seismic imaging with high fidelity. However, the source (forward) and receiver (backward) wavefields propagate along the opposite direction of time, and the recursive computation with the out-of-order access requires that all the wavefields of source propagation should be stored on the hard disk. For massive amounts of seismic data, saving the source wavefield from the central processing unit (CPU) [or graphics processing unit (GPU)] device to the disk and loading these data from the hard disk to the CPU (or GPU) device become extremely intensive in time and storage, which has been a bottleneck of Q-RTM. Several methods have been developed to reduce the huge wavefield storage in acoustic media, but are not applicable in the attenuated media. In this letter, we present a reversible hybrid absorbing boundary condition for Q-RTM, which is implemented by mixing the reversible attenuation and the random boundary conditions. Based on our developed new boundary, we just need to save the wavefield at the last one or two time steps in the forward process and then reconstruct the source wavefield in the time-reversal order. Numerical results demonstrate that the method can avoid the huge seismic data input and output (I/O) requirement and improve the computational efficiency dramatically.

Q-compensated reverse time migration in viscoacoustic media including surface topography

Conventional reverse time migration (RTM) may not produce high-quality images in areas with attenuation and severe topography because severe topographic surfaces have a great impact on seismic wave simulation, resulting in strong scattering and diffraction waves, and anelastic properties of the earth affect the kinematics and dynamics of seismic wave propagation. To overcome these problems, we have developed a [Formula: see text]-compensated topographic RTM method. In this method, a new viscoacoustic quasidifferential equation is introduced to simulate forward- and backward-propagated wavefields. The viscoacoustic equation has a lossy term and a dispersion term without memory variables, and it is solved by a hybrid spatial partial derivative scheme. A new stabilization operator is derived and substituted into the [Formula: see text]-compensated viscoacoustic quasidifferential equation to suppress high-frequency noise during the attenuated wavefield compensation. Numerical tests on a sag attenuating topographic model and an attenuating topographic Marmousi2 model demonstrate that our [Formula: see text]-compensated topographic RTM can produce accurate and high-quality images by correcting the anelastic amplitude loss and phase-dispersion effects. Finally, our method is tested on a field data set.

An efficient local operator-based Q-compensated reverse time migration algorithm with multistage optimization

Most existing [Formula: see text]-compensated reverse time migration ([Formula: see text]-RTM) algorithms are based on pseudospectral methods. Because of the global nature of pseudospectral operators, these methods are not ideal for efficient parallelization, implying that they may suffer from high computational cost and inefficient memory usage for large-scale industrial problems. In this work, we reported a novel [Formula: see text]-RTM algorithm — the multistage optimized [Formula: see text]-RTM method. This [Formula: see text]-RTM algorithm uses a finite-difference method to compensate the amplitude and the phase simultaneously by uniquely combining two techniques: (1) a negative [Formula: see text] method for amplitude compensation and (2) a multistage dispersion optimization technique for phase correction. To prevent high-frequency noise from growing exponentially and ruining the imaging results, we apply a finite impulse response low-pass filter using the Kaiser window. The theoretical analyses and numerical experiments demonstrate that this [Formula: see text]-RTM algorithm precisely recovers the decayed amplitude and corrects the distorted phase caused by seismic attenuation effects, and hence produces higher resolution subsurface images with the correct structural depth information. This new method performs best in the frequency range of 10–70 Hz. Compared with pseudospectral [Formula: see text]-RTM methods, this [Formula: see text]-RTM approach offers nearly identical imaging quality. Based on local numerical differential operators, this [Formula: see text]-RTM method is very suitable for parallel computing and graphic processing unit implementation, an important feature for large 3D seismic surveys.