Backward Wavefield Research Articles

Waveform inversion with structural regularizing constraint based on gradient decomposition

Abstract Full waveform inversion (FWI) can simultaneously update low-to-medium wavenumber velocity components and high-wavenumber velocity components. However, if seismic data lack large-offset data and effective low-frequency components, FWI updates will be dominated by high-wavenumber velocity perturbation. Meanwhile, providing that the initial model is inaccurate, inversion will have the problem of local minima. In this study, FWI is developed with structural regularizing constraint based on gradient decomposition (RGDFWI). By correlating the separated forward wavefield and backward wavefield with specific propagating direction, FWI gradient is decomposed into tomography-mode gradient and migration-mode gradient. We propose an optimized strategy taking full advantage of the two modes of FWI gradient. On the one hand, we use tomography-mode gradient to enhance low-to-medium wavenumber updates. On the other hand, we use migration-mode gradient to apply structural regularizing constraint by estimating structure dip and adding sparsity constraint in Seislet domain. During the inversion process, high-wavenumber structural information constrains and guides low-wavenumber model updates. The results of two numerical tests, Marmousi model test and Overthrust model test, validate the optimized strategy, which can produce a better initial velocity model for FWI. The inversion finally generates a high-precision and high-resolution velocity model.

Improving full-waveform inversion based on sparse regularization for geophysical data

Abstract Full-waveform inversion (FWI) is an advanced geophysical inversion technique. FWI provides images of subsurface structures with higher resolution in fields such as oil exploration and geology. The conventional algorithm minimizes the misfit error by calculating the least squares of the wavefield solutions between observed data and simulated data, followed by gradient direction and model update increment. Since the gradient is calculated by forward and backward wavefields, the high-accuracy model update relies on accurate forward and backward wavefield modelling. However, the quality of wavefield solutions obtained in practical situations could be poor and does not meet the requirements of high-resolution FWI. Specifically, the low-frequency wavefield is easily affected by noise and downsampling, which influences data quality, whereas the high-frequency wavefield is susceptible to spatial aliasing effects that produce imaging artefacts. Therefore, we propose using an algorithm called sparse relaxation regularized regression to optimize the wavefield solution in frequency-domain FWI, which is the forward and backward wavefield obtained from the Helmholtz equation, thus improving FWI's accuracy. The sparse relaxation regularized regression algorithm combines sparsity and regularization, allowing the broadband FWI to reduce the effects of noise and outliers, which can provide data supplementation in the low-frequency band and anti-aliasing in the high-frequency band. Our numerical examples demonstrate the wavefield optimization effect of the sparse relaxation regularized regression-based algorithm in various cases. The improved algorithm's accuracy and stability are verified compared to the Tikhonov regularization algorithm.

Crosswell frequency-domain reverse time migration imaging with wavefield decomposition

Abstract Crosswell seismic technique can provide high-resolution imaging and monitoring of the subsurface. However, compared to the surface seismic, crosswell data contain more complex wave components, which increases the difficulty of seismic processing and migrations. Conventional acoustic reverse time migration (RTM) mainly uses the cross-correlation of the source forward and the receiver backward wavefields. The redundant information generated by cross-correlation may undermine the imaging reliability in the crosswell models. Thus, we develop a novel wavefield decomposition imaging condition and only used cross-correlation information of incident and reflection waves in the same propagation directions, which eliminates the artifacts generated from the cross-correlation of wave information unrelated to reflections in the crosswell image. We perform RTM in the frequency domain to maintain efficiency in multi-shot problems. A mono-frequency wavefield decomposition method is applied to separate and process the seismic data. The forward and backward wavefields are reclassified into the up- and downpropagating components. The L1 norm is introduced to enhance the robustness of the proposed imaging method. We then use this method to synthesize data from layering models and analyze the imaging results generated from each pair of cross-correlations using source and receiver wavefields. Results show that the cross-correlation information belonging to the same propagation contributes most to the crosswell image, and the other information always generates migration noises. Moreover, we apply the proposed method to a real-field dataset. Processing results validate the effectiveness of the proposed means for eliminating false events in the crosswell models and improving image quality.

Viscoacoustic reverse time migration with stable and effective two-way attenuation compensation

Intrinsic attenuation in seismic wave propagation leads to amplitude dissipation and phase dispersion in seismic data. The attenuation (∼1/ Q) effect is a common cause for inaccurate image locations and dimmed image energy in reverse time migration (RTM). Conventional viscoacoustic or viscoelastic RTM methods implement attenuation compensation by reversing the sign of the amplitude loss term and keeping the dispersion term unchanged for forward and backward wavefield extrapolation, which requires the decoupled viscoacoustic wave equation and gives rise to the numerical instability issue. To address these problems, we develop a stable and effective two-way attenuation-compensated viscoacoustic RTM method. Our method uses a new two-way normalized crosscorrelation imaging condition to compensate the attenuation effect. In the new imaging condition, only the attenuated forward wavefield of sources and the viscoacoustic Green’s functions of receivers are required, which eliminates the instability source arising from the operation of reversing the sign of the dissipation term in the viscoacoustic wave equation. Because no modifications are made to the viscoacoustic wave equation, the new imaging condition can be calculated without any additional computational cost and can be applied to decoupled and coupled viscoacoustic wave equation-based RTMs, which is more flexible than other attenuation compensation strategies. Two synthetic tests and one field data example are presented to demonstrate the stability and effectiveness of our method.

A wavefield reconstruction method using sparse representation and dictionary learning for RTM

Abstract Reverse time migration (RTM) is a well-established imaging technique that uses the two-way wave equation to achieve high-resolution imaging of complex subsurface media. However, when using RTM for reverse time extrapolation, a source wavefield needs to be stored for cross-correlation with the backward wavefield. This requirement results in a significant storage burden on computer memory. This paper introduces a wavefield reconstruction method that combines sparse representation to compress a substantial amount of crucial information in the source wavefield. The method uses the K-SVD algorithm to train an adaptive dictionary, learned from a training dataset consisting of wavefield image patches. For each timestep, the source wavefield is divided into image patches, which are then transformed into a series of sparse coefficients using the trained dictionary via the batch-orthogonal matching pursuit algorithm, known for its accelerated sparse coding process. This novel method essentially attempts to transform the wavefield domain into the sparse domain to reduce the storage burden. We used several evaluation metrics to explore the impact of parameters on performance. We conducted numerical experiments using acoustic RTM and compared two RTM methods using checkpointing techniques with two strategies from our proposed method. Additionally, we extended the application of our method to elastic RTM. The conducted tests demonstrate that the method proposed in this paper can efficiently compress wavefield data, while considering both computational efficiency and reconstruction accuracy.

Isogeometric multi-resolution full waveform inversion based on the finite cell method

Full waveform inversion (FWI) is an iterative identification process that serves to minimize the misfit of model-based simulated and experimentally measured wave field data. Its goal is to identify a field of parameters for a given physical object. For many years, FWI is very successful in seismic imaging to deduce velocity models of the earth or of local geophysical exploration areas. FWI has also been successfully applied in various other fields, including non-destructive testing (NDT) and biomedical imaging. The inverse optimization process of FWI relies on forward and backward solutions of the (elastic or acoustic) wave equation, as well as on efficient computations of adequate optimization directions. Many approaches employ (low order) finite element or finite difference methods, using parameterized material fields whose resolution is chosen in relation to the elements or nodes of the discretized wave field. In our previous paper (Bürchner et al., 2023), we investigated the potential of using the finite cell method (FCM) as the wave field solver. The FCM offers the advantage that highly complex geometric models can be incorporated easily. Furthermore, we demonstrated that the identification of the model’s density outperforms that of the velocity — particularly in cases where unknown voids characterized by homogeneous Neumann boundary conditions need to be detected. The paper at hand extends this previous study in the following aspects: The isogeometric finite cell analysis (IGA-FCM) – a combination of isogeometric analysis (IGA) and FCM – is applied as the wave field solver, with the advantage that the polynomial degree and subsequently also the sampling frequency of the wave field can be increased quite easily. Since the inversion efficiency strongly depends on the accuracy of the forward and backward wave field solutions and of the gradients of the functional, consistent, and lumped mass matrix discretization are compared. The resolution of the grid describing the unknown material density – thus allowing to identify voids in a physical object – is then decoupled from the knot span grid. Finally, we propose an adaptive multi-resolution algorithm that locally refines the material grid using an image processing-based refinement indicator. The developed inversion framework allows fast and memory-efficient wave simulations and object identification. While we study the general behavior of the proposed approach using 2D benchmark problems, a final 3D problem shows that it can also be used to identify void regions in geometrically complex spatial structures.

Deep crustal structures with reverse time migration applied to offshore wide-angle seismic data: Equatorial and North-West Brazilian margins

We image the Moho discontinuity and deep crustal layers by applying the Reverse Time Migration (RTM) method to wide-angle seismic (WAS) data that were acquired along two different profiles in NW and SE offshore Brazil, where the spacing between OBS is 12.5 km and 150 m between shots. The application of this method is quite uncommon to ocean bottom seismometers (OBS) data due to the OBS wide spacing deployment and low folds. We analyze the effectiveness of the RTM method when applied to the reflectivity of the WAS data. We imaged the structures by cross-correlating the forward and backward wavefields given by the acoustic seismic equation. This allowed us to use each interface crossed by a ray as both a source and a receiver. The velocity models used to perform RTM were previously obtained by applying a procedure of two-dimensional forward ray-tracing followed by a damped least-squares travel time inversion. The results obtained have an unexpected large contribution from the wavefield traveling as refractions within the Earth and, because of that, we can talk about recovering refractors from the RTM results. We obtained strong and continuous refractors for depths of 7–15 km that correspond to the basement and the Moho discontinuity. As we move landwards, the refractors that correspond to the Moho discontinuity disappear due to the deepening of the surface. Finally, we investigate how the slight change on the velocity model influences the result of the RTM method. The obtained results are promising for a wide range of applications at a crustal scale seismic exploration, with wide-angle seismic data.

3C-3D tunnel seismic reverse time migration imaging: A case study of Pearl River Delta Water Resources Allocation Project

In recent years, tunnel boring machines (TBMs) have been widely used because of their fast construction speed and economical benefits, and the need to obtain geological information ahead of the tunnel to ensure the TBM tunnel construction safety has also arisen. Faced with the complex geological environment, seismic forward-prospecting is the primary method for predicting the unfavorable geological structure in front of the tunnel face. As a crucial step in seismic forward-prospecting method, migration can image geological characteristics and help learn the distribution of unfavorable geology. To achieve all-round imaging of unfavorable geology, we suggested a three-component (3C) three-dimensional (3D) tunnel seismic reverse time migration imaging method. The acoustic wave equation is employed for the forward and backward wavefield extrapolation after the P/S-wave separation to eliminate the imaging artifacts from severe converted waves in tunnels. GPU parallelization is used to accelerate the calculation to further adapt to the rapid excavation of TBMs. The proposed method is validated in two typical numerical models, especially for 3C seismic data that provides all-round geological information to achieve enhanced imaging energy and artifact reduction, which demonstrates the significance of 3C seismic data to migration imaging. This case study was conducted in the Pearl River Delta Water Resources Allocation Project in Guangdong Province, China. The reliability of 3C3D tunnel seismic reverse time migration imaging in field tunnel engineering was investigated. Two consecutive detections were performed, and then the spatial positions of faults and fractures in front of the tunnel face were accurately judged by the combination of the two imaging results. Based on the detection results, the excavation rate was slowed down and the tunnel lining support was increased. The TBM then successfully passed through the risk areas. This case study can serve as a reference for other tunnels with similar concerns regarding the necessity of 3C seismic data and consecutive detections.

A fast space-time-domain Gaussian beam migration approach using the dominant frequency approximation

The Gaussian beam migration (GBM) is a steady imaging approach, which has high accuracy and efficiency. Its implementation mainly includes the traditional frequency domain and the recent popular space-time domain. Firstly, we use the upward ray tracing strategy to get the backward wavefields. Then, we use the dominant frequency of the seismic data to simplify the imaginary traveltime calculation of the wavefields, which can cut down the Fourier transform number compared with the traditional GBM in the space-time domain. In addition, we choose an optimized parameter for the take-off angle increment of the up-going and down-going rays. These optimizations help us get an efficient space-time-domain acoustic GBM approach. Typical four examples show that the proposed method can significantly improve the computational efficiency up to one or even two orders of magnitude in different models with different model parameters and produce good imaging results with comparable accuracy and resolution with the traditional GBM in the space-time domain.

Three-dimensional Gauss–Newton constant-Q viscoelastic full-waveform inversion of near-surface seismic wavefields

SUMMARY Full-waveform inversion (FWI) methods rely on accurate numerical simulation of wave propagation in the analysed medium. Acoustic or elastic wave equations are often used to model seismic wave propagation. These types of simulations do not account for intrinsic attenuation effects due to material anelasticity, and thus correction techniques have been utilized in practice to partially compensate the anelasticity. These techniques often only consider the waveform amplitude correction based on averaging of overall amplitude response over the entire data set, and ignore the phase correction. Viscoelastic wave equations account for the anelastic response in both waveform amplitude and phase, and are therefore a more suitable alternative. In this study, we present a novel 3-D Gauss–Newton viscoelastic FWI (3-D GN-VFWI) method. To address the main challenge of the Gauss–Newton optimization, we develop formulas to compute the Jacobian efficiently by the convolution of virtual sources and backward wavefields. The virtual sources are obtained by directly differentiating the viscoelastic wave equations with respect to model parameters. In order to resolve complex 3-D structures with reasonable computational effort, a homogeneous attenuation (Q factor) is used throughout the analysis to model the anelastic effects. Synthetic and field experiments are performed to demonstrate the utility of the method. The synthetic results clearly demonstrate the ability of the method in characterizing a challenging velocity profile, including voids and reverse velocity layers. The field experimental results show that method successfully characterizes the complex substructure with two voids and undulating limestone bedrock, which are confirmed by invasive tests. Compared to 3-D elastic FWI results, the presented viscoelastic method produces more accurate results regarding depths of the voids and bedrock. This study suggests that the improvement of imaging accuracy would warrant the widespread use of viscoelastic wave equations in FWI problems. To our best knowledge, this is the first reported study on 3-D GN-VFWI at any scale. This study provides the new theory and formulation for the use of Gauss–Newton optimization on the 3-D viscoelastic problem.

Efficient least-squares reverse time migration using local cross-correlation imaging condition

Abstract The data-domain least-squares reverse time migration (LSRTM), a promising imaging method for obtaining a high-resolution reflectivity image, can be implemented by matching the de-migrated data to the observed one. However, LSRTM requires expensive computational costs in its implementation due to tremendous memory usage (direct storage on disk/memory) for saving source wavefields and repeatable forward/backward wavefield simulations with iterations compared to conventional RTM. Although the source wavefield reconstruction technique can be used to reduce the memory usage, additional computational cost is inevitable because it is implemented during backward wavefield simulation. To alleviate the computational burden, we have developed an efficient LSRTM scheme with local cross-correlation imaging condition (LSRTM-LC), which can use pre-saved source wavefields according to the window size. Because the window size is much shorter than the total recording time for wavefield extrapolation, storing source wavefields into computer memory can be feasible. In addition, the procedure for storing source wavefields is implemented only at the first iteration because the background velocity is fixed during iterations of LSRTM. To validate the feasibility of the LSRTM-LC scheme, we have carried out several numerical tests with synthetic datasets. Numerical tests demonstrated that LSRTM-LC can provide us with equivalent de-migrated data from saved source wavefields and the gradient vector obtained from LSRTM-LC compared to those obtained from the conventional LSRTM. As a result, LSRTM-LC can generate the same quality of reflectivity models as conventional LSRTM, however, it requires less computational costs compared to conventional LSRTM.

Generating surface-offset common-image gathers with backward wavefield synthesis

Surface-offset common-image gathers (CIGs) are an important data format for seismic velocity analysis. However, the reverse time migration (RTM) method, which is wavefield propagation based, does not directly produce surface-offset CIGs because it propagates waves from all the offsets together. Here, we implement the generation of surface-offset CIGs by synthesizing the backward wavefields using the forward wavefields at locations where source and receiver locations overlap. This method produces surface-offset CIGs with high accuracy and low cost when compared with those generated by backpropagating each trace separately. We adopt nonnegative least-squares filters for sparse linear deconvolution. The computational effectiveness of the proposed method increases for higher dimensions, higher-order stencils, and more complicated wave equations. The proposed method works stably on a realistic towed-streamer acquisition system with moderate geometric positioning errors between the locations of airguns and hydrophones.

Viscoacoustic reverse time migration with a robust space-wavenumber domain attenuation compensation operator

Intrinsic attenuation gives rise to phase dispersion and amplitude loss during seismic wave propagation. Not correcting these effects in seismic imaging can result in inaccurate reflector locations, dimmed amplitudes, and degraded spatial resolution. In reverse time migration (RTM), attenuation compensation can be implemented by reversing the sign of the dissipation term and keeping the dispersion term unchanged for backward wavefield extrapolation. Although this Q-compensated RTM scheme can effectively correct attenuation effects, amplitude amplification during backpropagation might lead to numerical instabilities, especially for field data with strong high-frequency noise. To mitigate this problem, we have developed a robust space-wavenumber compensation operator and applied it to viscoacoustic RTM. By analyzing the dispersion-only and viscoacoustic Green’s functions, we obtain an analytical solution for the attenuation compensation operator in a homogeneous medium. Because it is a time-frequency operator, to apply it directly in viscoacoustic RTM requires access to the extrapolated wavefields within a certain time window. To avoid storing the wavefields and improve the computational efficiency, we use an approximated dispersion relation and convert the time-frequency operator to an equivalent space-wavenumber operator, which allows us to implement attenuation compensation on the fly during wavefield extrapolation. The hybrid-domain property of the operator enables us to account for the wavenumber-dependent compensation. A similar strategy also can be applied to the migrated images as a poststack processing approach, which is more efficient than the prestack compensation. Two synthetic and one land field data set examples demonstrate the feasibility and adaptability of our method.

Three-dimensional passive-source anisotropic reverse time migration for imaging lithospheric discontinuities

SUMMARY The scattered teleseismic body waves have been used intensively to characterize the receiver-side lithospheric structures. The routinely used ray-theory-based methods have limitations to image complex structures and tackle strong heterogeneities. The newly developed wave-equation based, passive-source reverse time migration (RTM) approach can overcome such limitations. To date, passive-source RTM has been developed only for isotropic media. However, at least to the first-order, most lithospheric structures possess effective transverse isotropy with spatially variable symmetry directions. It is important to know the magnitudes of imaging errors that may be introduced by passive-source RTM when seismic anisotropy is treated in an incorrect way. In this paper, we investigate the influence of elastic anisotropy on teleseismic P-to-S conversions at the lithospheric discontinuities and explain why an isotropic RTM may fail to focus the converted wavefields from the perspective of relative arrival time variations with backazimuth and shear wave splitting. Accordingly, we extend the passive-source RTM approach for imaging 3-D lithospheric targets possessing transverse isotropy from the following two aspects: First, the teleseismic recordings with direct P and converted S phases are reverse-time extrapolated using rotated staggered grid (RSG) pseudospectral method, which can tackle strong heterogeneity and transverse isotropies with symmetry axes in arbitrary direction; Secondly, the backward elastic wavefields are efficiently decomposed into vector anisotropic P and S modes to support accurate imaging.

A reverse time migration-based multistep angular spectrum approach for ultrasonic imaging of specimens with irregular surfaces

We develop a new ultrasonic imaging framework for non-destructive testing of an immersed specimen featuring an irregular top surface and demonstrate its capability of accurately depicting the lower surfaces of multiple damages hidden in the specimen. Central to the framework is a multistep angular spectrum approach (ASA), via which the forward propagation wavefields of wave sources and backward propagation wavefields of the received wave signals are calculated. Upon applying a zero-lag cross-correlation imaging condition of reverse time migration (RTM) to the obtained forward and backward wavefields, the image of the specimen with an irregular surface can be reconstructed, in which hidden damages, if any and regardless of quantity, are visualized. The effectiveness and accuracy of the framework are examined using numerical simulation, followed with experiment, in both of which multiple side-drilled holes, at different locations in aluminum blocks with various irregular surfaces, are characterized. Results have proven that multiple damages in a specimen with an irregular surface can be individually localized, and the lower surface of each damage can further be imaged accurately, thanks to the RTM-based algorithm in which multiple wave reflections from the specimen bottom are taken into wavefield extrapolation. The proposed imaging approach presents higher computational efficiency, compared to conventional RTM, and enhanced imaging contrast over prevailing total focusing methods.

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.

An efficient vector elastic reverse time migration method in the hybrid time and frequency domain for anisotropic media

Vector elastic reverse time migration (ERTM) produces subsurface elastic images with correct polarities using multicomponent seismic data. However, the decomposition of elastic wavefields into vector P- and S-wavefields is computationally expensive, particularly in heterogeneous and complex anisotropic media. We have developed a computationally efficient vector ERTM method in the hybrid time and frequency domain by combining three existing techniques. Rather than decomposing elastic wavefields into vector qP- and qS-wavefields during time-domain wavefield propagation, we conduct the wavefield decomposition in the frequency domain for several selected frequencies. In general, the number of selected frequencies needed for migration imaging is much smaller than the number of time steps during forward and backward wavefield propagation, leading to greatly reduced computational costs associated with the qP-/qS-wavefield vector separation in complex heterogeneous anisotropic media. We further combine an implicit directional wavefield separation into the vector ERTM to enhance the image quality. The numerical results demonstrate that our method produces high-quality elastic-wave migration images with notably reduced computational costs compared to the conventional vector ERTM method.

Stable and High-Efficiency Attenuation Compensation in Reverse-Time Migration Using Wavefield Decomposition Algorithm

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.

Least-squares reverse time migration using analytic-signal-based wavefield decomposition

An intrinsic problem during migration and imaging of seismic wavefields using the two-way wave equation is the crosstalk interference between the up/down propagation of the corresponding source and receiver wavefields. To mitigate this crosstalk, the downgoing source and upgoing receiver wavefield imaging condition (IC) is adopted at an early stage of the inversion process, improving convergence and obtaining cleaner reflection images. A wavefield decomposition methodology can also be incorporated into a least-squares reverse time migration (LSRTM) algorithm. The separation of wavefields based on the propagation direction in the early iterations of LSRTM is to reduce interference noise during the inversion process given that the IC considers only primary reflections. Wavefields decomposed with respect to the vertical direction can be easily obtained by Fourier transforms on the time and vertical axes; however, they usually require significantly higher computational effort especially for 3D applications. Vertical wavefield decomposition by a complex-valued analytic signal is an alternative method implemented by the Hilbert transform, which can be conducted by 1D Fourier transform only on the vertical axis. An LSRTM algorithm adopting this decomposition method has a disadvantage in that it requires two additional wave modelings at each iteration. However, by adapting the deprimary IC into LSRTM, only one more modeling is additionally required in the backward wavefield propagation as compared with conventional LSRTM. Our LSRTM using wavefield decomposition has the ability to produce broader band reflectivity images than conventional LSRTM. This is demonstrated with numerical examples using synthetic and real data resulting artifact-free migration results and broadband reflectivity images.

Reverse time migration via frequency-adaptive multiscale spatial grids

Reverse time migration (RTM) is widely used because of its ability to recover complex geologic structures. However, RTM also has a drawback in that it requires significant computational cost. In RTM, wave modeling accounts for the largest part of the computing cost for calculating forward- and backward-propagated wavefields before applying an imaging condition. For this reason, we have applied a frequency-adaptive multiscale spatial grid to enhance the efficiency of the wave simulations. To implement wave modeling for different values of the spatial grid interval, we apply a model reduction technique, the generalized multiscale finite-element method (GMsFEM), which solves local spectral problems on a fine grid to simulate wave propagation on a coarser grid. We can enhance the speed of computation without sacrificing accuracy by using coarser grids for lower frequency waves, while applying a finer grid for higher frequency waves. In the proposed method, we can control the size of the coarse grid and level of heterogeneity of the wave solutions to tune the trade-off between speedup and accuracy. As we increase the expected level of complexity of the wave solutions, the GMsFEM wave modeling can capture more detailed features of waves. After computing the forward and backward wavefield on the coarse grid, we reproject the coarse wave solutions to the fine grid to construct the RTM gradient image. Although wave solutions are computed on a coarse grid, we still obtain the RTM images without reducing the image resolution by projecting coarse wave solutions to the fine grid. We determine the efficiency of the proposed imaging method using the Marmousi-2 model. We compare the RTM images using GMsFEM with a fixed coarse mesh and a multiple frequency-adaptive coarse meshes to indicate the image quality and computational speed of the new approach.