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Reverse-time-migration artifacts occur when diving waves, head waves or backscattered waves crosscorrelate. These events are particularly strong where high velocity contrasts exist. Simple filtering of the final image can lead to good results but might compromise the integrity of the signal of interest. We demonstrate that a better technique is to apply least-squares filtering with prediction-error filters, a method traditionally used for S/N separation.

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... Moreover, low-cut filters can reduce the amplitude of steeply dipping reflectors, which mainly contain low-frequency information. Guitton et al. (2006) presented another possible post-migration solution by applying a least-squares filter to RTM images. In addition, the application of a Laplacian filter with appropriate pre-and post-migration processing is an effective solution that can remove RTM artifacts without hurting steeply dipping reflectors (Zhang and Sun, 2009). ...

... The computation of the wavefield propagation direction at every point in the compuational domain can be done using a Poynting vector (Yoon et al., 2004). This works well for simple models, but it does not produce satisfactory results in complex subsurface structures (Guitton et al., 2006). Xie and Wu (2006) applied the local one-way propagator, which is endowed with the Rayleigh integral, in order to decompose the wavefields. ...

Reverse-time migration (RTM) is capable of imaging very steeply dipping reflectors and overhangs. However, it usually produces strong artifacts that contaminate the shallow parts of the migrated images. These artifacts can be suppressed using an imaging condition with the decomposed source and receiver wavefields corresponding to backscattered events. This imaging condition keeps only energy at the points where strong backscattering originates with respect to Cartesian directions. In this paper, such a technique is applied and examined. The results show that RTM using wavefield decomposition is a promising remedy for attenuating artifacts compared to the implementation of a low-cut filter. However, some artifacts still remain in the decomposed RTM image. These residual arti-facts are caused by cross-correlation of backscattered waves and the overlapping portions of decomposed wavefields with opposite propagation directions with respect to Cartesian directions. Such overlap results from the FFT of discontinuous decomposed wavefields in the fourier domain.

... Several implementations of RTM using the cross-correlation imaging condition have been reported [5], [6], [7], [8], however, this imaging condition often produces a significant number of strong amplitudes and low-frequency noise that contaminates the model. This lowfrequency noise (artifacts) results from singularities in the velocity field (strong velocity contrasts) and unwanted cross-correlation of source and receiver wavefields in non-reflective points along the ray-paths. ...

... These low-frequency artifacts are not present in one-way equationbased migration models built with the same cross-correlation imaging condition. Several works have been developed around to attenuate these low-frequency artifacts, preserving reflections, and improving model quality, implementing other strategies such as modifications of the wave equation [9], [10], proposing other imaging conditions [3], [11]- [15], and using image filtering techniques [6], [8]. ...

Low-frequency artifacts in reverse time migration result from unwanted cross-correlation of the source and receiver wavefields at non-reflecting points along ray-paths. These artifacts can hide important details in migrated models and increase poor interpretation risk.
Some methods have been proposed to avoid or reduce the number of these artifacts, preserving reflections, and improving model quality, implementing other strategies such as modification of the wave equation, proposing other imaging conditions, and using image filtering techniques. One of these methods uses wavefield decomposition, correlating components of the wavefields that propagate in opposite directions.
We propose a method for extracting directional information from the RTM imaging condition wavefields to obtain characteristics allowing for better, more refined imaging. The method works by separating directional information about the wavefields based on the continuous wavelet transform (CWT), and the analysis of the main changes on the frequency content revealed within the scalogram obtained by a Gaussian wavelet family.
Through numerical applications, we demonstrate that this method can effectively remove undesired artifacts in migrated images. In addition, we use the Laguerre-Gauss filtering to improve the results obtained with the proposed method.

... Moreover, low-cut filters can reduce the amplitude of steeply dipping reflectors, which mainly contain low-frequency information. Guitton et al. (2006) presented another possible post-migration solution by applying a least-squares filter to RTM images. In addition, the application of a Laplacian filter with appropriate pre-and post-migration processing is an effective solution that can remove RTM artifacts without hurting steeply dipping reflectors (Zhang and Sun, 2009). ...

... The computation of the wavefield propagation direction at every point in the compuational domain can be done using a Poynting vector (Yoon et al., 2004). This works well for simple models, but it does not produce satisfactory results in complex subsurface structures (Guitton et al., 2006). Xie and Wu (2006) applied the local oneway propagator, which is endowed with the Rayleigh integral, in order to decompose the wavefields. ...

Reverse-time migration (RTM) is capable of imaging very steeply dipping reflectors and overhangs. However, it usually produces strong artifacts that contaminate the shallow parts of the migrated images. These artifacts can be suppressed using an imaging condition with appropriate decomposed source and receiver wavefields. In this paper, such a technique is applied and examined. This imaging condition keeps only energy at the points where strong backscattering originates. The results show that RTM using wavefield decomposition is a promising remedy for attenuating artifacts compared to the implementation of a low-cut filter. However, some artifacts still remain in the decomposed RTM image. These residual artifacts are caused by the cross-correlation between the upgoing component of the direct source wavefield and the backscattered component of the receiver wavefield.

... The process of migration returns the underground reflection point information back to properly positioned reflections, and the reflection wave, simultaneously with the diffraction wave, automatically converges and interferes with automatic decomposition; better migration methods can provide high resolution interpretation of imaging. Compared with Kirchhoff migration method [25,26], reverse time migration (RTM) can effectively use the full wave field information [27][28][29]. While compared with one-way wave equation migration [30], RTM has no limitations in propagation direction and dip angle, since RTM does not need to separate the wave field, and can better use the reversed branch and the multiple waves; it is quite adaptive to lateral velocity variations, too. ...

The distribution of the permafrost in the Tibetan Plateau has dramatically changed due to climate change, expressed as increasing permafrost degradation, thawing depth deepening and disappearance of island permafrost. These changes have serious impacts on the local ecological environment and the stability of engineering infrastructures. Ground penetrating radar (GPR) is used to detect permafrost active layer depth, the upper limit of permafrost and the thawing of permafrost with the season’s changes. Due to the influence of complex structure in the permafrost layer, it is difficult to effectively characterize the accurate structure within the permafrost on the radar profile. In order to get the high resolution GPR profile in the Tibetan Plateau, the reverse time migration (RTM) imaging method was applied to GPR real data. In this paper, RTM algorithm is proven to be correct through the groove’s model of forward modeling data. In the Beiluhe region, the imaging result of GPR RTM profiles show that the RTM of GPR makes use of diffracted energy to properly position the reflections caused by the gravels, pebbles, cobbles and small discontinuities. It can accurately determine the depth of the active layer bottom interface in the migration section. In order to prove the accuracy of interpretation results of real data RTM section, we set up the three dielectric constant models based on the real data RTM profiles and geological information, and obtained the model data RTM profiles, which can prove the accuracy of interpretation results of three-line RTM profiles. The results of three-line RTM bears great significance for the study of complex structure and freezing and thawing process of permafrost at the Beiluhe region on the Tibetan Plateau.

... Least-squares migration (LSM), which can be regarded as a linearized waveform inversion method (Tarantola, 1984;Duan et al., 2016;Chen and Sacchi, 2017;Gu et al., 2018), attempts to find the inverse of modeling via repeated migrations and demigrations (Tarantola, 1984;Lambaré et al., 1992;Schuster, 2017). Therefore, LSM can effectively improve the illumination and resolution of the underground (Guitton et al., 2006;Dai et al., 2011;Yuan et al., 2017;Zhu et al., 2018). Given a smooth background velocity, a velocity perturbation can be obtained by LSM through minimizing the objective function (Nemeth et al., 1999;Ren et al., 2016;Gu et al., 2017). ...

Least-squares reverse time migration (LSRTM), an effective tool for imaging the structures of the earth from seismograms, can be characterized as a linearized waveform inversion problem. We have investigated the performance of three minimization functionals as the 𝐿2 norm, the hybrid 𝐿1/𝐿2 norm, and the Wasserstein metric (𝑊1 metric) for LSRTM. The 𝑊1 metric used in this study is based on the dynamic formulation of transport problems, and a primal-dual hybrid gradient algorithm is introduced to efficiently compute the 𝑊1 metric between two seismograms. One-dimensional signal analysis has demonstrated that the 𝑊1 metric behaves like the 𝐿1 norm for two amplitude-varied signals. Unlike the 𝐿1 norm, the 𝑊1 metric does not suffer from the differentiability issue for null residuals. Numerical examples of the application of three misfit functions to LSRTM on synthetic data have demonstrated that, compared to the 𝐿2 norm, the hybrid 𝐿1/𝐿2 norm and 𝑊1 metric can accelerate LSRTM and are less sensitive to non-Gaussian noise. For the field data application, the 𝑊1 metric produces the most reliable imaging results. The hybrid 𝐿1/𝐿2 norm requires tedious trial-and-error tests for the judicious threshold parameter selection. Hence, the more automatic 𝑊1 metric is recommended as a robust alternative to the customary 𝐿2 norm for time-domain LSRTM.

... The gradient of the LS-RTM objective function that we derived in the previous sections uses the zero-lag crosscorrelation imaging condition and maps seismic reflections in the observed data to a perturbation in the medium parameters, which are in this case the velocity in squared slowness (s 2 km −2 ). One of the well-known shortcomings of imaging velocity perturbations with the zero-lag crosscorrelation imaging condition are low-frequency imaging artifacts that result from backscattering of the source wavefield (e.g., Yoon and Marfurt, 2006;Guitton et al., 2007). This issue is especially problematic for imaging salt bodies because high velocity contrasts in the migration velocity model lead to reflections/ backscattering of the downgoing wavefield, thus creating strong low-frequency artifacts in the image (Figure 1). ...

Least-squares reverse time migration is a powerful approach for true-amplitude seismic imaging of complex geologic structures, but the successful application of this method is currently hindered by its enormous computational cost, as well as its high memory requirements for computing the gradient of the objective function. We have tackled these problems by introducing an algorithm for low-cost sparsity-promoting least-squares migration using on-the-fly Fourier transforms. We formulate the least-squares migration objective function in the frequency domain (FD) and compute gradients for randomized subsets of shot records and frequencies, thus significantly reducing data movement and the number of overall wave equations solves. By using on-the-fly Fourier transforms, we can compute an arbitrary number of monochromatic FD wavefields with a time-domain (TD) modeling code, instead of having to solve individual Helmholtz equations for each frequency, which becomes computationally infeasible when moving to high frequencies. Our numerical examples demonstrate that compressive imaging with on-the-fly Fourier transforms provides a fast and memory-efficient alternative to TD imaging with optimal checkpointing, whose memory requirements for a fixed background model and source wavelet are independent of the number of time steps. Instead, the memory and additional computational costs grow with the number of frequencies and determine the amount of subsampling artifacts and crosstalk. In contrast to optimal checkpointing, this offers the possibility to trade the memory and computational costs for image quality or a larger number of iterations and is advantageous in new computing environments such as the cloud, where computing is often cheaper than memory and data movement.

... There have been many studies that address low-wavenumber migration noise, including the use of Poynting vectors (Yoon and Marfurt, 2006), modifying wave equations with directional damping terms (Fletcher et al., 2006), least-squares filtering (Guitton et al., 2006), and Laplacian filtering (Zhang and Sun, 2009). Liu et al. (2011) introduce a cost-efficient solution based on Hilbert transforms over depth; the cross-correlation imaging condition effectively applies only for source and receiver wavefields along opposite directions. ...

Elastic reverse time migration (E-RTM) using full wave equation is able to generate both P-P and converted P-S images of the subsurface. Besides the ability of imaging complicated
geological structures, similar to acoustic imaging, image artifacts are also observed in the P-P and P-S images, where high velocity contrasts/gradients in the velocity model produce reflections of the source and receiver wavefields. Separating RTM images into different components based on wave propagation directions is able to effectively remove such image artifacts. We first compare different image separation methods, including up/down image separation during, and post, imaging. The post-imaging method is simple, as it requires only simple operations applied to the stacked RTM images. The separated up/down images are similar to those separated during imaging, but there are also difference. The difference comes from different physical meanings of the up/down separated images. Furthermore, we present a new workflow that combines a cost-efficient up/down imaging condition and a post-imaging separation method. A numerical example using the Sigsbee 2A model indicates that the up/down separated P-P and P-S images from the new workflow are equivalent to those by duringimaging separation methods.

... In this case, cross-correlation of these two wavefields will generate some noise in the gradient. This issue in conventional RTM is well addressed ( Guitton et al., 2006). However, the standard gradient formula used in LSRTM has some differences with the cross-correlation imaging condition (Chattopadhyay and McMechan, 2008) normally used in conventional RTM. ...

We propose to apply a novel incoherent dictionary learning (IDL) algorithm for regularizing the least-squares inversion in seismic imaging. The IDL is proposed to overcome the drawback of traditional dictionary learning algorithm in losing partial texture information. Firstly, the noisy image is divided into overlapped image patches, and some random patches are extracted for dictionary learning. Then, we apply the IDL technology to minimize the coherency between atoms during dictionary learning. Finally, the sparse representation problem is solved by a sparse coding algorithm, and image is restored by those sparse coefficients. By reducing the correlation among atoms, it is possible to preserve most of the small-scale features in the image while removing much of the long-wavelength noise. The application of the IDL method to regularization of seismic images from least-squares reverse time migration shows successful performance.

... (Tarantola, 1984;Pratt et al., 1998;Brossier et al., 2009;Virieux and Operto, 2009). Although some weaknesses exist in RTM, such as the huge memory cost and the low-frequency artifacts, researchers are seeking to reduce these weak points with skillful techniques (Yoon and Marfurt, 2006;Guitton et al., 2007;Clapp, 2009). ...

Conventional migration uses the seismic data set recorded at a given depth as one initial condition from which to implement wavefield extrapolation in the depth domain. In using only one initial condition to solve the second-order acoustic wave equation, some approximations are used, resulting in the limitation of imaging angles and inaccurate imaging amplitudes.We use an over/under bilayer sensor seismic data acquisition system that can provide the two initial conditions required to make the second-order acoustic wave equation solvable in the depth domain, and we develop a two-way wave equation depth migration algorithm by adopting concepts from one-way propagators, called bilayer sensor migration. In this new migration method, two-way wave depth extrapolation can be achieved with two one-way propagators by combining the wavefields at two different depths. It makes it possible to integrate the advantages of one-way migration methods into the bilayer sensor system. More detailed bilayer sensor migration methods are proposed to demonstrate the feasibility. In the impulse response tests, the propagating angle of the bilayer sensor migration method can reach up to 90°, which is superior to those of the corresponding one-way propagators. To test the performance, several migration methods are used to image the salt model, including the one-way generalized screen propagator, reverse time migration (RTM), and our bilayer sensor migration methods. Bilayer sensor migration methods are capable of imaging steeply dipping structures, unlike one-way propagators; meanwhile, bilayer sensor migration methods can greatly reduce the numbers of artifacts generated by salt multiples in RTM.

... One of the well known shortcomings of imaging velocity perturbations with the zero-lag cross-correlation imaging condition, are low frequency imaging artifacts that result from backscattering of the source wavefield (e.g. Yoon and Marfurt, 2006;Guitton et al., 2007). This issue is especially problematic for imaging salt bodies, as high velocity contrasts in the migration velocity model lead to reflections/backscattering of the down-going wavefield, thus creating strong low-frequency artifacts in the image (Figure 1). ...

Least-squares reverse-time migration is a powerful approach for true amplitude seismic imaging of complex geological structures, but the successful application of this method is currently hindered by its enormous computational cost, as well as high memory requirements for computing the gradient of the objective function. We tackle these problems by introducing an algorithm for low-cost sparsity-promoting least-squares migration using on-the-fly Fourier transforms. We formulate the least-squares migration objective function in the frequency domain and compute gradients for randomized subsets of shot records and frequencies, thus significantly reducing data movement and the number of overall wave equations solves. By using on-the-fly Fourier transforms, we can compute an arbitrary number of monochromatic frequency-domain wavefields with a time-domain modeling code, instead of having to solve individual Helmholtz equations for each frequency, which becomes computa-tionally infeasible when moving to high frequencies. Our numerical examples demonstrate that compressive imaging with on-the-fly Fourier transforms provides a fast and memory-efficient alternative to time-domain imaging with optimal checkpointing, whose memory requirements for a fixed background model and source wavelet is independent of the number of time steps. Instead, memory and additional computational cost grow with the number of frequencies and determine the amount of subsampling artifacts and crosstalk. In contrast to optimal checkpointing, this offers the possibility to trade both memory and computational cost for image quality or a larger number of iterations and is advantageous in new computing environments such as the cloud, where compute is often cheaper than memory and data movement.

... On the other hand, the proposed imaging functions require the local orientation of the interface to be known at least approximately a priori, which POLLITZ ELASTIC IMAGING OF MATERIAL INTERFACES 2874 10.1029/2018JB017089 constitutes a disadvantage. They also do not address the artifacts that can arise in RTM when strong velocity gradients or reflectors are present in the reference velocity model (e.g., Denli & Huang, 2008;Guitton et al., 2007;Liu et al., 2011;Yoon & Marfurt, 2006;Youn & Zhou, 2001), a problem recently addressed using an imaging condition based on the elastic energy norm (Rocha et al., 2016). Examples with 2-D synthetic structures based on a 2.5-D spectral element method (SEM) are intended to highlight the properties of the proposed reflectivity kernels, test their ability to recover the location and sense of material property contrasts, and enable the comparison of volumetric sensitivity kernels with the reflectivity kernels. ...

Reverse time migration is often used to interpret acoustic or three-component seismic recordings by creating an image of subsurface seismic reflectors. Here I describe elastic reverse time migration imaging functions that are cast as waveform misfit sensitivity kernels of contrasts in material parameters across hypothetical seismic discontinuities, that is, specular reflectors. The proposed “surface” imaging functions are theoretically applicable to either reflected or converted waves in order to estimate the location and reflectivity of these discontinuities. The surface imaging functions, as well as volumetric sensitivity kernels that target point diffractors, are tested on sets of synthetic surface array recordings that sample the 3-D seismic wavefield on simple 2-D structures generated using a 2.5-D spectral element method. These tests illustrate that in contrast with the volumetric sensitivity kernels, the reflectivity is generally dominated by a high-amplitude peak that coincides with input locations of discontinuities. Passive recordings of microseismicity, shot gathers, or a combination thereof can be potentially interpreted with the new surface imaging functions to yield useful reflectivity images. Published 2019. This article is a U.S. Government work and is in the public domain in the USA.

... We investigate the effect of two types of preconditioners in LSRTM. The first preconditioner is a 2D Laplacian operator, which has the property of filtering the low wavenumber components (Guitton et al., 2006). In this case, the equation (10) takes the form ...

Least-squares reverse time migration (LSRTM) aims to improve the quality of seismic images by fitting the reflection wavefield with a linearized inverse scattering model, using conjugate gradient (CG) iterations. Due its high computational cost, it is important to accelerate the convergence rate in LSRTM. With this objective we investigate a preconditioner for LSRTM inspired by the asymptotic inverse scattering expression for RTM. The proposed preconditioner can significantly reduce in the number of CG iterations in LSRTM than the more conventional laplacian preconditioner.

... This problem is widely reported in the literature (e.g. Yoon and Marfurt (2006); Guitton et al. (2007)). To remove these imaging artifacts, we replace the conventional imaging condition for RTM by the inverse-scattering imaging condition (Stolk et al., 2012;Whitmore and Crawley, 2012;Witte et al., 2017). ...

Least-squares reverse time migration is well-known for its capability to generate artifact-free true-amplitude subsurface images through fitting observed data in the least-squares sense. However, when applied to realistic imaging problems, this approach is faced with issues related to overfitting and excessive computational costs induced by many wave-equation solves. The fact that the source function is unknown complicates this situation even further. Motivated by recent results in stochastic optimization and transform-domain sparsity-promotion, we demonstrate that the computational costs of inversion can be reduced significantly while avoiding imaging artifacts and restoring amplitudes. While powerful, these new approaches do require accurate information on the source-time function, which is often lacking. Without this information, the imaging quality deteriorates rapidly. We address this issue by presenting an approach where the source-time function is estimated on the fly through a technique known as variable projection. Aside from introducing negligible computational overhead, the proposed method is shown to perform well on imaging problems with noisy data and problems that involve complex settings such as salt. In either case, the presented method produces high resolution high-amplitude fidelity images including an estimates for the source-time function. In addition, due to its use of stochastic optimization, we arrive at these images at roughly one to two times the cost of conventional reverse time migration involving all data.

... This approach is efficient and effective as it acts like a high-pass filter which attenuates many of the low-frequency artifacts. Guitton et al. (2007a) demonstrated a least-squares prediction error filter to remove artifacts based on S/N separation. ...

Reverse time migration (RTM) is a seismic imaging method to map the subsurface reflectivity using recorded seismic waveforms. The practice in exploration seismology has long established a two-fold approach of seismic imaging: Using velocity modeling building to establish the long-wavelength reference velocity models, and using seismic migration to map the short-wavelength reflectivity structures. Among various seismic migration methods for different situations, RTM is the only method that is capable to use all seismic wave types that can be computed numerically. Being initiated in early 1980's, RTM seeks an image of the subsurface reflectivity as the best match in an image space between the extrapolation of time-reversed waveform data and the prediction based on estimated velocity model and source parameters. Judging the image quality in the same space of forming the images is more advantageous than the approaches of modeling and inversion which seek the solution in the model space but judge its fitness in data space. Considering that most seismic migration applications today still use primary reflection as the only signal, the capability of RTM to use all computable wave types is unique and helpful reducing the imaging artifacts due to mistaking non-primary waves as primary reflections. Hence, we refer to those RTM algorithms using only primary reflections as the first-generation RTM methods, and the RTM algorithms making a full use of primary reflections, multiple reflections and other non-primary waveform data as the second-generation RTM methods. This paper reviews the development history of the RTM along with its major challenges, current solutions, and future perspectives.

... The vertical derivative filter acting over the image is designed to remove the low-wavenumber artifact (Guitton et al., 2007). The blue and red line for PP and PS wave, respectively. ...

... There have been many studies that address lowwavenumber migration noise, including the use of Poynting vectors (Yoon and Marfurt 2006), modifying wave equations with directional damping terms (Fletcher et al. 2006), least-squares filtering (Guitton et al. 2006), and Laplacian filtering (Zhang and Sun 2009). Liu et al. (2011) introduce a cost-efficient solution based on Hilbert transforms over depth; the conventional cross-correlation imaging condition effectively applies only when the source and receiver wavefields are of opposite propagation directions (e.g., the down-going component of the source wavefield and the up-going component of the receiver wavefield). ...

Reverse time migration (RTM) using a two-way wave equation is able to generate high-quality images of the subsurface. Besides the ability to image complicated geological features, image artifacts can be observed in elastic P and S wave images, where high velocity contrasts/gradients in the velocity model produce back-scattering reflections during wavefield extrapolations. Separating RTM images into components based on wavefield propagation directions is able to effectively remove such image artifacts. We first compare image separation methods, including up/down image separation during, and post, imaging. The post-imaging method is easy to implement using existing RTM software, as it requires only simple operations applied to the stacked RTM images. The separated up/down images are similar to those separated during imaging, but there are also differences. The differences come from the different physical meanings of the up/down separated images. The during-imaging methods separate RTM image into its up/down components using propagation directions of the individual source/receiver wavefields, while the post-imaging method separates images based on the vector sum of the source and receiver wavefield directions. Inspired by these differences, we develop a composite workflow that combines a cost-efficient up/down imaging condition and a post-imaging separation step, which provides an alternative solution for obtaining up/down separated images from full waveform imaging. Numerical example using an elastic version of the Sigsbee 2A model indicates that the up/down separated images from the proposed workflow are effectively equivalent to those obtained using the during-imaging separation methods. More interestingly, we show the pros and cons in the imaging results from different up/down separation methods, especially in converted P–S images.

... Fletcher et al. (2005aFletcher et al. ( , 2005b introduce a directional damping factor into the wave equation to suppress unwanted reflection. Guitton et al. (2007) use a least-squares attenuation method that is not able to attenuate the artifacts completely. Yoon and Marfurt (2006) present a Poynting-vector imaging condition that was used variously by several researchers to suppress the artifacts (Costa et al., 2009;Du and Qin, 2009;Araujo et al., 2014;Chen and Huang, 2014;Jin et al., 2014). ...

Reverse time migration (RTM) as a full wave equation method can image steeply dipping structures incorporating all waves without dip limitation. It causes a set of low-frequency artifacts that start to appear for reflection angles larger than 60°. These artifacts are known as the major concern in RTM method. We are first to attempt to formulate a scheme called the leapfrog-rapid expansion method to extrapolate the wavefields and their first derivatives. We have evaluated a new imaging condition, based on the Poynting vectors, to suppress the RTM artifacts. The Poynting vectors information is used to separate the wavefields to their downgoing and upgoing components that form the first part of our imaging condition. The Poynting vector information is also used to calculate the reflection angles as a basis for our weighting function as the second part of the aforementioned imaging condition. Actually, the weighting function is applied to have the most likely desired information and to suppress the artifacts for the angle range of 61°-90°. This is achieved by dividing the angle range to a triplet domain from 61° to 70°, 71° to 80°, and 81° to 90°, where each part has the weight of cos(θ), cos3/2(θ), and cos²(θ), respectively. It is interesting to note that, besides suppressing the artifacts, the weighting function also has the capability to preserve crosscorrelation from the real reflecting points in the angle range of 61°-90°. Finally, we tested the new RTM procedure by the BP synthetic model and a real data set for the North Sea. The obtained results indicate the efficiency of the procedure to suppress the RTM artifacts in producing high-quality, highly illuminated depth-migrated image including all steeply dipping geologic structures.

... This problem is widely reported in the literature (e.g. Yoon and Marfurt (2006); Guitton et al. (2007)). To remove these imaging artifacts, we replace the conventional imaging condition for RTM by the inverse-scattering imaging condition (Stolk et al., 2012;Whitmore and Crawley, 2012;Witte et al., 2017). ...

Least‐squares reverse time migration is well‐known for its capability to generate artifact‐free true‐amplitude subsurface images through fitting observed data in the least‐squares sense. However, when applied to realistic imaging problems, this approach is faced with issues related to overfitting and excessive computational costs induced by many wave‐equation solves. The fact that the source function is unknown complicates this situation even further. Motivated by recent results in stochastic optimization and transform‐domain sparsity‐promotion, we demonstrate that the computational costs of inversion can be reduced significantly while avoiding imaging artifacts and restoring amplitudes. While powerfull, these new approaches do require accurate information on the source‐time function, which is often lacking. Without this information, the imaging quality deteriorates rapidly. We address this issue by presenting an approach where the source‐time function is estimated on the fly through a technique known as variable projection. Aside from introducing negligible computational overhead, the proposed method is shown to perform well on imaging problems with noisy data and problems that involve complex settings such as salt. In either case, the presented method produces high resolution high‐amplitude fidelity images including an estimates for the source‐time function. In addition, due to its use of stochastic optimization, we arrive at these images at roughly one to two times the cost of conventional reverse time migration involving all data.
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... Filtering of wavefields is commonly done using Poynting vectors (Yoon and Marfurt, 2004) in order to separate down-and up-going events before cross-correlation (Pestana et al., 2013). In image filtering methods, such as least-squares artifact attenuation (Guitton et al., 2007) or Laplacian preconditioning (Zhang and Sun, 2009), the artifacts are removed in the image itself after cross-correlating the full source and receiver wavefields. An alternative imaging condition called linearized inverse scattering imaging condition which addresses low-frequency RTM artifacts was introduced by Op't Root et al. (2012). ...

... The denoising methods in the field of seismic imaging can be divided into three groups: (1) denoising during wave propagation (Baysal et al. 1983;Loewenthal et al. 1987;Fletcher et al. 2005), (2) applying imaging conditions (Yoon et al. 2004;Yoon & Marfurt 2006;Liu et al. 2007;Bulcão et al. 2007;Costa et al. 2009;Nguyen & McMechan 2013), and (3) denoising after processing (Yoon & Marfurt 2006;Karazincir & Gerrard 2006;Guitton & Biondi 2007;Zhang & Sun 2009). The objective of after-processing is the final imaging result rather than the process of wavefield extrapolation or imaging conditions; thus, it is easy and convenient to realise as well as adaptive to complex media. ...

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.

... Unlike Kirchhoff migration and one-way wave-equation migration, reverse time migration (RTM), using the two-way wave equation, has higher imaging accuracy and no dip limitation. RTM has become a state-of-the-art technique for imaging complex subsurface structures such as salt flanks (McMechan, 1983;Zhu and Lines, 1998;Guitton et al., 2007;Zhang and Sun, 2009;Liu et al., 2011). RTM migrates the receiver wavefield along all directions and propagation paths and thus suffers from low-wavenumber artifacts. ...

Reverse time migration (RTM) generally uses the zero-lag crosscorrelation imaging condition, requiring the source and receiver wavefields to be known at the same time step. However, the receiver wavefield is calculated in time-reversed order, opposite to the order of the forward-propagated source wavefield. The inconvenience can be resolved by storing the source wavefield on a computer memory/disk or by reconstructing the source wavefield on the fly for multiplication with the receiver wavefield. The storage requirements for the former approach can be very large. Hence, we have followed the latter route and developed an efficient source wavefield reconstruction method. During forward propagation, the boundary wavefields at N layers of the spatial grid points and a linear combination of wavefields at M − N layers of the spatial grid points are stored. During backward propagation, it reconstructs the source wavefield using the saved wavefields based on a new finite-difference stencil ( M is the operator length parameter, and 0 ≤ N ≤ M). Unlike existing methods, our method allows a trade-off between accuracy and storage by adjusting N. A maximum-norm-based objective function is constructed to optimize the reconstruction coefficients based on the minimax approximation using the Remez exchange algorithm. Dispersion and stability analyses reveal that our method is more accurate and marginally less stable than the method that requires storage of a combination of boundary wavefields. Our method has been applied to 3D RTM on synthetic and field data. Numerical examples indicate that our method with N = 1 can produce images that are close to those obtained using a conventional method of storing M layers of boundary wavefields. The memory usage of our method is ( N + 1)/ M times that of the conventional method.

... Compared with the DSO misfit function in Shen et al. (2003), an additional vertical derivative is applied to remove the low-wavenumber artifacts in the ERTM result (Guitton et al., 2007;Weibull and Arntsen, 2013). ...

Multicomponent seismic data acquisition can reveal more information about geologic structures and rock properties than single component acquisition. Full elastic wave seismic imaging, which uses multicomponent seismic to its full potential, is promising because it provides more opportunities to understand the material properties of the earth by the joint use of P- and S-waves. A prerequisite of seismic imaging is the availability of a reliable macrovelocity model. Migration velocity analysis for P-waves, which can fill that requirement for the P-wave velocity, has been well-studied, especially under the acoustic approximation. However, a reliable estimation of the S-wave velocities remains troublesome. Elastic wave-equation migration velocity analysis has the potential to build P- and S-wave velocity models together, but it inevitably suffers from the effects of mode coupling and conversion in the forward and adjoint wavefield reconstructions. We have developed a differential semblance optimization approach to sequentially invert the background P- and S-wave velocity models from extended PP- and PS-images in the subsurface offset domain. Preconditioning of the gradients with respect to the S-wave velocity through mode decoupling can improve the reliability of the optimization. Numerical investigations with synthetic examples demonstrate the effectiveness of gradient preconditioning and the feasibility of our migration velocity analysis approach for elastic wave imaging.

... The image restoration processing such as de-noising and deblurring of radar profiles is one of the basic tasks in the GPR data processing. Guitton et al. (2006) pointed out the shortcomings of the traditional filtering method to change the phase and spectrum information of the imaging results, and realized the least square filtering. Zhang and Sun (2009) used the Laplacian filtering method to improve the imaging resolution. ...

Reverse time migration (RTM) is the important intermediate step for focusing the radar diffracted energy of the targets in ground penetrating radar imaging. The conventional RTM algorithm demands a large number of iterative trial experiments and depends on the experts' decision on the estimation of the velocity or relative permittivity of the subsurface medium. Meanwhile, the RTM profile is vulnerable to artifacts, which are composed of noise interference, multiple interferences, arc-shaped clutter, and crosstalk, so it is difficult to inspect visually. Therefore, we propose a RTM method based on accurate velocity estimation and total variation (TV) de-noising to improve the accuracy of the RTM imaging. Firstly, the appropriate migration velocity is obtained automatically by autofocusing metrics to reduce the number of visual inspection times and corrections in the migration processing. Secondly, the TV de-noising strategy based on split Bregman iterative is applied to the RTM profile with the cross-correlation imaging condition, so that the edge of the target can be obvious and the position can be accurate. Then, we apply the proposed method to the simulation data of the pipeline model and the tunnel lining model. All results show that the selected three different autofocusing metrics have unimodality and unbiasedness, which can focus on a single relative permittivity to obtain appropriate migration velocity. Furthermore, the TV de-noising strategy successfully eliminates artifacts, reconstructs contours, enhances the edge sharpness, and improves the quality and accuracy of the radar profile. Finally, we take the field data of the LiuYang River tunnel to verify the applicability of our method, we choose 2–30 as a wider range of relative permittivity based on prior information. Considering the different lateral velocity of the field data, we adopt lateral segmentation processing to improve the quality of the GPR profile. The tests of simulation data and the field data indicate that the proposed method can provide a scientific and effective way for accurate interpretation of GPR data.

... Baysal et al., 1984;Loewenthal et al., 1987;Fletcher et al., 2005); the second is to improve the imaging conditions by using the Poynting vector (e.g. Yoon and Marfurt, 2006) and decomposing wavefields into up-going and down-going ones (Liu et al., 2011a), etc; the third is to apply post-imaging filtering, such as Laplacian filtering, least-squares filtering (Guitton et al., 2007), dip filtering (Sun and Zhang, 2009), etc. ...

... Imaging condition approaches to remove the effects of such multiples include using the Poynting vector (Richardson and Malcolm, 2014), deconvolution (Valenciano and Biondi, 2003) and local slopes (Sava, 2007). Post imaging approaches include filtering common image gathers (cigs) (Biondi and Shan, 2002) or filtering the final image (Youn and Zhou, 2001;Guitton et al., 2007). Multiple prediction methods may also be used to identify multiples in the migrated section to aid interpretation. ...

The ability to synthesize recordings from surface data as if they had come from subsurface sources has allowed geophysicists to estimate subsurface properties. Either in the form of classical seismic migration which creates structural maps of the subsurface, to the more recent seismic interferometry which turns seismic sources into receivers and vice-versa, this ability has provided a rich trove of methods with which to probe the Earth's interior. While powerful, both of these techniques suffer from well-known issues. Standard migration requires data without multiply-scattered waves (multiples). Seismic interferometry, on the other hand, can be applied to full recorded data (containing multiples and other wave types), but requires sources (receivers) to be physically placed at the location from (to) one wishes to estimate responses. The Marchenko method, developed recently for the seismic setting, circumvents both of these restrictions: it creates responses from virtual subsurface sources as if measured at the surface. It requires only single-sided surface data, and a smooth estimate of the subsurface velocities. Initially developed for acoustic media, this thesis contributes the first elastic formulation of the Marchenko method, providing a more suitable setting for applications for the solid Earth. In another development, this thesis shows how the obtained virtual recordings may be used for migration. With these two contributions, this thesis shows that for elastic surface seismic data, the main drawbacks of migration and interferometry can be overcome using the Marchenko method: multiples do not harm migrated images, and sources (receivers) need not be physically placed in the medium for their responses to be accessible. In addition to the above methods, generating images devoid of multiple-related artifacts can be achieved in several other different ways. Two approaches to this are the use of a post-imaging filter, and attenuation of internal multiples in the data itself. This thesis contributes one new method using each of these approaches. First, a form of Marchenko imaging is known to create spurious reflectors, as also occurs in standard reverse-time migration (RTM). However, these artifacts usually appear at different locations in RTM and this form of Marchenko imaging. Using this insight, this thesis presents a way to combine pairs of seismic images in such a way that their differences (e.g. artifacts) are attenuated, while similarities (e.g. true reflectors) are preserved. Applying this to RTM and Marchenko-derived images markedly improves image quality. Second, this thesis presents a method to estimate multiples in the data. Multiples can either be migrated on their own to aid in interpretation, or be adaptatively removed from the data to improve image quality. However, because of the nature of adaptive subtraction, this second method may harm primary energy. To avoid this problem, this thesis develops a final method to directly image using only primary energy in the recorded data using only a small number of virtual points. This method bypasses the need for multiple removal and the estimation of subsurface responses at every depth location. In addition, primaries from particular reflectors may be particularly selected such that they can be imaged individually. Overall this thesis provides several new ways to use surface seismic data in such a way that multiples do not hamper the end product of seismic data processing: the seismic image. It demonstrates this use on synthetic and real data, proving their effectiveness.

Seismic velocity estimation is a challenging task, especially when no initial model is present. In most cases, a traveltime tomography approach is used as a significant part of the workflow. However, it requires noise-sensitive, time-consuming picking and uses a ray approximation of the wave equation. Time reversal (TR) is a fundamental physical concept, based on the wave equation's invariance under TR operation. If the recorded wavefield is reversed and back-propagated into the medium, it will focus at its original source location regardless of the complexity of the medium. We use this property for seismic velocity analysis, formulated as an inversion problem with focusing at the known source location and onset time as the objective function. It is globally solved using competitive particle swarm optimization and an adequate model parameterization. This approach has the advantages of using the wave equation, being picking-free, handling low signal-to-noise ratio and requiring neither information on the source wavelet nor an initial velocity model. Although the method is discussed in the framework of direct source-receiver path acquisition, the foundations for its use with conventional reflection data are laid. We have determined the method's usefulness and limitations using synthetic and field crosshole acquisition examples. In both cases, inversion results are compared with a standard traveltime tomography approach and illustrate the advantages of using TR focusing.

In this work the analytical wavefield is computed by just solving the wave equation once, differently of conventional methods that need to solve the wave equation twice: once for the source and another for the Hilbert transformed source. Our proposed method can improve the computation of wavefield separation and can bring the causal imaging condition into practice. For time extrapolation, we are using the rapid expansion method to compute the wavefield and its first order time derivative and then compute the analytical wavefield. This method is unconditionally stable and free of numerical noise. By computing the analytical wavefield, we can, therefore, separate the wavefield into down- and up-going components for each time step in an explicit way. For RTM applications, we can now employ the causal imaging condition and through a synthetic example, we could demonstrate the effectiveness of this new imaging condition without applying a Laplacian filter. The RTM result shows that it can successfully remove the low-frequency noise which is common in the typical cross-correlation imaging condition.

In this work the analytical wavefield is extrapolated in time using the one-step extrapolated exponential matrix method. The method is based on the solution of a first order differential wave equation and it employs the Chebyshev polynomial expansion to approximate the exponential of the matrix and besides of being unconditional stable, it propagates waves free of numerical dispersion noise and it is able to extrapolate waves in time using a time step up to Nyquist's limit. Using an analytical wavefield we can therefore separate the wavefield into up- and down- going source and receiver wavefields for each time step in an explicit way. Employing a causal imaging condition, through a synthetic example, we demonstrate the effectiveness of this new imaging condition without applying a Laplacian filter. The RTM result shows that it can successfully remove the low-frequency noise which is common in the traditional cross-correlation imaging condition.

Conventional seismic migration and inversion are inherently limited in their ability to detect and characterize subsurface elements smaller than the seismic wavelength, such as faults, pinchouts, karsts, fractures, fluid contact, etc. However, those elements, playing an important role in seismic exploration and production, act as scattering objects, which can be effectively detected and positioned using the time reversal (TR) principle. We use TR to spatially localize subsurface sources in passive seismic scenarios and scatterers in active seismic surveys, both having the physical properties of a point diffractor. The method uses numerical back propagation of the time-reversed registered wavefield followed by an analysis of its obtained focusing, based on a supervised learning approach. In this novel approach, no imaging condition is applied. In addition, it does not require knowledge of the source wavelet and it accounts for multiple scattering. The usefulness of the method is demonstrated using synthetic and field examples.

In areas with strong velocity gradients, traditional reverse‐time migration (RTM) based on cross‐correlation imaging condition not only produce low‐frequency noise but also generate diving wave artefacts. The artefacts caused by diving waves have no typical low‐frequency characteristics and cannot be eliminated by simple high‐pass filtering approaches. We apply the wave‐field decomposition imaging condition to analyse the causes of false images in RTM by decomposing the full wave‐field into up‐going and down‐going components in the angle domain. We find that artificial diving wave imaging artefacts, which are generated by the cross‐correlation between the up‐going source and down‐going receiver wave fields in areas with strong velocity gradients, arise at large angles. We propose an efficient strategy by means of the wavelength‐dependent smoothing operator to eliminate artefacts from artificial diving waves in RTM. Specifically, the proposed method provides more reasonable down‐going wave‐fields in areas with sharp velocity constructs by considering the factor of varying seismic wavelengths during wave propagation, and the artificial components of diving waves are eliminated in a straightforward manner. Meanwhile, the other wave‐field components that contribute to true subsurface images are minimally affected. Benefiting from a smoothed velocity, the proposed method can be adapted to the traditional RTM imaging frame, which reveals significant implementation potential for the seismic exploration industry. A salt model is designed and included to demonstrate the effectiveness of our approach.
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The separation of up- and downgoing wavefields is an important technique in the processing of multicomponent recorded data, propagating wavefields, and reverse time migration (RTM). Most of the previous methods for separating up/down propagating wavefields can be grouped according to their implementation strategy: a requirement to save time steps to perform Fourier transform over time or construction of the analytical wavefield through a solution of the wave equation twice (one for the source and another for the Hilbert-transformed source), in which both strategies have a high computational cost. For computing the analytical wavefield, we are proposing an alternative method based on the first-order partial equation in time and by just solving the wave equation once. Our strategy improves the computation of wavefield separation, and it can bring the causal imaging condition into practice. For time extrapolation, we are using the rapid expansion method to compute the wavefield and its first-order time derivative and then we can compute the analytical wavefield. By computing the analytical wavefield, we can, therefore, separate the wavefield into up- and downgoing components for each time step in an explicit way. Applications to synthetic models indicate that our method allows performing the wavefield decomposition similarly to the conventional method, as well as a potential application for the 3D case. For RTM applications, we can now use the causal imaging condition for several synthetic examples. Acoustic RTM up/down decomposition demonstrates that it can successfully remove the low-frequency noise, which is common in the typical crosscorrelation imaging condition, and it is usually removed by applying a Laplacian filter. Moreover, our method is efficient in terms of computational time when compared to RTM using an analytical wavefield computed by two propagations, and it is a little more costly than conventional RTM using the crosscorrelation imaging condition.

We describe a new method, full waveform inversion by model extension (FWIME) that recovers accurate acoustic subsurface velocity models from seismic data, when conventional methods fail. We leverage the advantageous convergence properties of wave-equation migration velocity analysis (WEMVA) with the accuracy and high-resolution nature of acoustic full waveform inversion (FWI) by combining them into a robust mathematically-consistent workflow with minimal need for user inputs. The novelty of FWIME resides in the design of a new cost function using the variable projection method, and a novel optimization strategy to combine the two techniques, making our approach more efficient and powerful than applying them sequentially. We observe that FWIME mitigates the need for accurate initial models and low-frequency long-offset data, which can be challenging to acquire. We generate three cycle-skipped 2D synthetic datasets, each containing only one type of wave (transmitted, reflected, refracted), and we analyze how FWIME successfully recovers accurate solutions with the same procedure for all three cases. In a second paper, we apply FWIME to challenging realistic examples where we simultaneously invert all wave modes.

This paper presents a reverse time migration (RTM) method formulated as the transpose of the forward operator. For modeling, the wave equation solution is expressed by the rapid expansion method (REM). The REM is a wave equation solution method that is based on the Chebyshev expansion and can be used to stably extrapolate wavefields even for larger time steps. The forward operator is commonly reused in RTM for back-propagation and achieves satisfactory results, but, in order to correctly apply the reverse time migration, it requires the adjoint wave-equation solution. Here, we show that the adjoint operator using the REM as the forward modeling operator can be obtained by transposing the forward operator. The new adjoint operator based on the REM is easily implemented with little changes in the existing RTM code. During the imaging condition procedure, we choose the causal imaging condition which is employed to avoid low-frequency noise and false events produced by the conventional cross-correlation imaging condition. A numerical example is used to compare the results produced by the traditional RTM and the proposed reverse adjoint time method and also to show the benefits of the adjoint method versus the conventional RTM.

Straightforward implementations of elastic reverse time migration (ERTM) often produce imaging artifacts associated with incorrectly imaged mode conversions, cross-talk, and back-scattered energies. To address these issues, we present three approaches, 1) vector-based normalized cross-correlation imaging conditions (VBNICs), 2) directional separation of wavefields to remove low-wavenumber noise, and 3) post-imaging filtering of the dip-angle gathers to eliminate the artifacts caused by non-physical wave modes. These approaches are combined to create an effective ERTM workflow which can produce high-quality images. Numerical examples demonstrate that, first, VBNICs can produce correct polarities for PP/PS images and can compute migrated dip-angle gathers efficiently by using P/S decomposed Poynting vectors. Second, they achieve improved signal-to-noise and higher resolution when performing up/down decomposition before applying VBNICs, and left/right decomposition enhances steep dips imaging at the computational cost of adding the Hilbert transform to a spatial direction. Third, dip filtering using slope-consistency analysis attenuates the remaining artifacts effectively. An application of the SEAM model demonstrates that the proposed ERTM workflow reduces noise and improves imaging ability for complex geological areas.

Reverse time migration is an advanced seismic migration imaging method. When the source wavefield and the receiver wavefield are cross‐correlated, the cross‐correlations of direct arrivals, backscattered waves and overturned waves will produce a lot of low‐frequency noise, which will mask the final imaging results. Laplacian filtering, as a common method to suppress low‐frequency noise, can adapt to any complex media, just adding a little computational cost. However, simple direct Laplacian filtering will destroy the characteristics of the useful signals. Therefore, the amplitude needs to be compensated before filtering when using the Laplace filtering method. Zhang and Sun (2009) proposed an improved Laplacian filtering method and gave a simple calculation formula and explanation. This method can effectively suppress the low‐frequency noise in reverse time migration while retaining the useful signal characteristics, but lacks detailed and strict mathematical derivation. Therefore, this paper gives a detailed and rigorous mathematical derivation of the amplitude‐compensated Laplace filtering from the point of view of amplitude‐preserved filtering. The source wavelet is used instead of the source wavefield to compensate amplitude, just adding a little calculation cost. Finally, the amplitude‐compensated Laplace filtering method is verified by two theoretical models and compared with the direct Laplacian filtering method. This article is protected by copyright. All rights reserved

Seismic imaging and parameter estimation are an import class of inverse problems with practical relevance in resource exploration, carbon control and monitoring systems for geohazards. The goal of seismic inverse problems is to image subsurface geological structures and estimate physical rock properties such as wave speed or density. Mathematically, this can be achieved by solving an optimization problem in which we minimize the mismatch between numerically modeled data and observed data from a seismic survey. As wave propagation through a medium is described by wave equations, seismic inverse problems involve solving a large number of partial differential equations (PDEs) during numerical optimization using finite difference modeling, making them computationally expensive. Additionally, seismic inverse problems are typically ill-posed, non-convex or ill-conditioned, thus making them challenging from a mathematical standpoint as well. Similar to the field of deep learning, this calls for software that is not only optimized for performance, but also enables geophysical domain specialists to experiment with algorithms in high-level programming languages and using different computing environments, such as high-performance computing (HPC) clusters or the cloud. Furthermore, they call for the adaption of dimensionality reduction techniques and stochastic algorithms to address computational cost from the algorithmic side. This thesis makes three distinct contributions to address computational challenges encountered in seismic inverse problems and to facilitate algorithmic development in this field. Part one introduces a large-scale framework for seismic modeling and inversion based on the paradigm of separation of concerns, which combines a user interface based on domain specific abstractions with a Python package for automatic code generation to solve the underlying PDEs. The modular code structure makes it possible to manage the complexity of a seismic inversion code, while matrix-free linear operators and data containers enable the implementation of algorithms in a fashion that closely resembles the underlying mathematical notation. The second contribution of this thesis is an algorithm for seismic imaging, that addresses its high computational cost and large memory imprint through a combination of on-the-fly Fourier transforms, stochastic sampling techniques and sparsity-promoting optimization. The algorithm combines the
best of both time- and frequency-domain inversion, as the memory imprint is independent
of the number of modeled time steps, while time-to-frequency conversions avoid the need
to solve Helmholtz equations, which involve inverting ill-conditioned matrices. Part three
of this thesis introduces a novel approach for adapting the cloud for high-performance
computing applications like seismic imaging, which does not rely on a fixed cluster of
permanently running virtual machines. Instead, computational resources are automatically
started and terminated by the cloud environment during runtime and the workflow takes
advantage of cloud-native technologies such as event-driven computations and containerized batch processing. The performance and cost analysis shows that this approach is able to address current shortcomings of the cloud such as inferior resilience, while at the same time reducing operating cost up to an order of magnitude. As such, the workflow provides a strategy for cost effectively running large-scale seismic imaging problems in the cloud and is a viable alternative to conventional HPC clusters.

Reverse time migration (RTM) is widely used in the industry because of its ability to handle complex geologic models including steeply dipping interfaces. The quality of images produced by RTM is significantly influenced by the performance of the numerical methods used to simulate the wavefields. Recently, a weighted Runge-Kutta discontinuous Galerkin (WRKDG) method has been developed to solve the wave equation, which is stable, explicit and efficient in parallelization and suppressing numerical dispersion. By incorporating two different weights for the time discretization, we obtain a more stable method with larger maximal Courant numbers. We apply this numerical method to RTM to handle complex topography and improve the imaging quality. By comparing it with high-order Lax-Wendroff correction (LWC) method, we show WRKDG is efficient in RTM. From the results of the Sigsbee2B data, we can find that our method is efficient in suppressing artifacts and can produce images of better quality when coarse meshes are used. The RTM results of the Canadian Foothills model also prove its ability in handling complex geometry and rugged topography.

Using staggered-grid finite difference method to solve seismic wave equation, large spatial grid and high dominant frequency of source cause numerical dispersion, staggered-grid finite difference method, which can reduce the step spatial size and increase the order of difference, will multiply the calculation amount and reduce the efficiency of solving wave equationThe optimal nearly analytic discrete (ONAD) method can accurately solve the wave equation by using the combination of displacement and gradient of spatial nodes to approach the spatial partial derivative under rough grid and high-frequency condition. In this study, the ONAD method is introduced into the field of reverse-time migration (RTM) for performing forward- and reverse-time extrapolation of a two-dimensional acoustic equation, and the RTM based on ONAD method is realized via normalized cross-correlation imaging condition, effectively suppressed the numerical dispersion and improved the imaging accuracy. Using ONAD method to image the groove model and SEG/EAGE salt dome model by RTM, and comparing with the migration sections obtained by staggered-grid finite difference method with the same time order 2nd and space order 4th, results show that the RTM based on ONAD method can effectively suppress numerical dispersion caused by the high frequency components in source and shot records, and archive accurate imaging of complex geological structures especially the fine structure, and the migration sections of the measured data show that ONAD method has practical application value.

Prestack depth migration is a key technology for imaging of complex reservoirs in media with strong lateral velocity variations. Prestack migrations are broadly separated into ray‐based and wave‐equation‐based methods. Because of its efficiency and flexibility, ray‐based Kirchhoff migration is popular in the industry. However, it has difficulties in dealing with the multi‐arrivals, caustics and shadow zones. On the other hand, wave‐equation‐based methods produce images superior to that of the ray‐based methods, but they are expensive numerically, especially methods based on two‐way propagators in imaging large regions. Therefore, reverse time migration algorithms with Gaussian beams have recently been proposed to reduce the cost, as they combine the high computational efficiency of Gaussian beam migration and the high accuracy of reverse time migration. However, this method was based on the assumption that the subsurface is isotropic. As the acquired azimuth and maximum offsets increase, taking into account the influence of anisotropy on seismic migration is becoming more and more crucial. Using anisotropic ray tracing systems in terms of phase velocity, we proposed an anisotropic reverse time migration using the Gaussian beams method. We consider the influence of anisotropy in the propagation direction and calculate the amplitude of Gaussian beams with optimized correlation coefficients in dynamic ray tracing, which simplifies the calculations and improves the applicability of the proposed method. Numerical tests on anisotropic models demonstrate the efficiency and accuracy of the proposed method, which can be used to image complex structures in presence of anisotropy in the overburden. This article is protected by copyright. All rights reserved

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.

Least-squares reverse time migration (LSRTM) overcomes the shortcomings of conventional migration algorithms by iteratively fitting the demigrated synthetic data and the input data to refine the initial depth image toward true reflectivity. It gradually enhances the effective signals and removes the migration artifacts such as swing noise during conventional migration. When imaging the subsalt area with complex structures, many practical issues have to be considered to ensure the convergence of the inversion. We tackle those practical issues such as an unknown source wavelet, inaccurate migration velocity, and slow convergence to make LSRTM applicable to subsalt imaging in geologic complex areas such as the Gulf of Mexico. Dynamic warping is used to realign the modeled and input data to compensate for minor velocity errors in the subsalt sediments. A windowed crosscorrelation based confidence level is used to control the quality of the residual computation. The confidence level is further used as an inverse weighting to precondition the data residual so that the convergence rates in shallow and deep images are automatically balanced. It also helps suppress the strong artifacts related to the salt boundary. The efficiency of the LSRTM is improved so that interpretable images in the area of interest can be obtained in only a few iterations. After removing the artifacts near the salt body using LSRTM, the image better represents the true geology than the outcome of conventional RTM, thus it facilitates the interpretation. Synthetic and field data examples are given to examine and demonstrate the effectiveness of the adaptive strategies. © 2017 Society of Exploration Geophysicists and American Association of Petroleum Geologists.

Depth imaging with multiples is a prestack depth migration method that uses multiples as the signal for more accurate boundary mapping and amplitude recovery. The idea is partially related to model-based multiple-suppression techniques and reverse-time depth migration. Conventional reverse-time migration uses the two-way wave equation for the backward wave propagation of recorded seismic traces and ray tracing or the eikonal equation for the forward traveltime computation (the excitation-time imaging principle). Consequently, reverse-time migration differs little from most other one-way wave equation or ray-tracing migration methods which expect only primary reflection events. Because it is almost impossible to attenuate multiples without degrading primaries, there has been a compelling need to devise a tool to use multiples constructively in data processing rather than attempting to destroy them. Furthermore, multiples and other nonreflecting wave types can enhance boundary imaging and amplitude recovery if a full two-way wave equation is used for migration. The new approach solves the two-way wave equation for both forward and backward directions of wave propagation using a finite-difference technique. Thus, it handles all types of acoustic waves such as reflection (primary and multiples), refraction, diffraction, transmission, and any combination of these waves. During the imaging process, all these different types of wavefields collapse at the boundaries where they are generated or altered. The process goes through four main steps. First, a source function (wavelet) marches forward using the full two-way scalar wave equation from a source location toward all directions. Second, the recorded traces in a shot gather march backward using the full two-way scalar wave equation from all receiver points in the gather toward all directions. Third, the two forward- and backward-propagated wavefields are correlated and summed for all time indicts. And fourth, a Laplacian image reconstruction operator is applied to the correlated image frame. This technique can be applied to all types of seismic data: surface seismic, vertical seismic profile (VSP), crosswell seismic, vertical cable seismic, ocean-bottom cable (OBC) seismic, etc. Because it migrates all wave types, the input data require no or minimal preprocessing (demultiple should not be done, but near-surface or acquisition-related problems might need to be corrected). Hence, it is only a one-step process from the raw field gathers to a final depth image. External noise in the raw data will not correlate with the forward wavefield except for some coincidental matching; is usually unnecessary to do signal enhancement processing before the depth imaging with multiples. The input velocity model could be acquired from various methods such as iterative focusing analysis or tomography as in other prestack depth migration methods. The new method has been applied to data sets from a simple multiple-generating model, the Marmousi model, and a real offset VSP. The results show accurate imaging of primaries and multiples with overall significant improvements over conventionally imaged sections.

In seismic modeling and in migration it is often desirable to use a wave equation (with varying velocity but constant density) which does not produce interlayer reverberations. The conventional approach has been to use a one-way wave equation which allows energy to propagte in one dominant direction only, typically this direction being upward or downward (Claerbout, 1972). The authors introduce a two-way wave equation which gives highly reduced reflection coefficients for transmission across material boundaries. For homogeneous regions of space, however, this wave equation becomes identical to the full acoustic wave equation. Possible applications of this wave equation for forward modeling and for migration are illustrated with simple models.

A significant improvement of seismic image resolution is obt ained by framing the shot- profile migration imaging condition as a 2-D deconvolution i n the shot position/time (xs, t ) domain. This imaging condition gives a better image resolution than the crosscorrelation imaging condition and is more stable than the "more conventional" 1-D deconvolution imaging condition. A resolution increment is also observed in common image gathers (CIGs) computed with the 2-D deconvolution imaging condition, thus allowing a more accurate velocity analysis.

SUMMARY We derive a new generalized imaging condition based on time shifts between source and receiver wavefields. This imaging condition con- trasts with other imaging techniques requiring space shifts between the two wavefields. This imaging condition is applicable to both Kirchhoff and wave-equation migrations. The transformation allows us to gener- ate common-image gathers presented as function of either time-shift or pseudo-angle at every location in space. Inaccurate migration velocity is revealed by common-image gathers with non-flat events.

Migration of stacked or zero-offset sections is based on deriving the wave amplitude in space from wave field observations at the surface. Conventionally this calculation has been carried out through a depth extrapolation. We examine the alternative of carrying out the migration through a reverse time extrapolation. This approach may offer improvements over existing migration methods, especially in cases of steeply dipping structures with strong velocity contrasts. This migration method is tested using appropriate synthetic data sets. -Authors

Two‐way migration methods require significantly greater computational resources than one‐way migration methods. However, their advantage is that they can not only handle multiarrivals, but have virtually no dip limitation, enabling imaging of overturned reflections. Reverse time migration, a wave equation technique using two‐way propagation, correctly handles both multiarrivals and the phase changes due to caustics, but by using two‐way propagation, does not suffer from dip limitation like one‐way downward continuation techniques. Although one‐way wave equation techniques can be implemented and classified as true‐amplitude migrations, reverse time migration makes no approximations that must be corrected to control amplitude.
Undesired reflections from interfaces in the velocity model are easily suppressed in poststack reverse time migration by forcing acoustic impedance to be constant. This paper focuses on suppressing such artifacts in prestack reverse time migration with a zero‐lag cross correlation imaging condition by applying a directional damping term to the non‐reflecting wave equation during propagation.

Thesis (Ph. D.)--University of Houston, 1998. Degree granted by Dept. of Geosciences. Includes bibliographical references.

SUMMARY We present a simple method for computing angle-domain Common Image Gathers (CIGs) using prestack reverse time migration. The proposed method is an extension of the method proposed by Rickett and Sava (2001) to compute CIGs by downward-continuation shot-profile migration. We demonstrate with a synthetic example the use of the CIG gathers for migration velocity updating. A challenge for imaging both overturned and prismatic reflections is the discrimination of the reflection generated on either side of interfaces. We show how the propagation direction of the reflections can be determined by evaluating the crosscorrelation of the source wavefield with the receiver wavefield at time lags different than zero. Reflections can be easily separated once their direction of propagation is determined. We demonstrate the method by imaging overturned events generated by a segment of dipping reflector immersed in a vertically layered medium. We also applied the method to a North Sea data set with overturned events. The results of reverse time prestack migration are superior to the one obtained by a downward-continuation migration, and the CIGs obtained by applying the proposed method provide useful information for velocity updating.

Migration of an observed zero-offset wavefield can be performed as the solution of a boundary value problem in which the data are extrapolated backward in time. This concept is implemented through a finite-difference solution of the two-dimensional acoustic wave equation. All depths are imaged simultaneously at time 0 (the imaging condition), and all dips (right up to vertical) are correctly migrated. Numerical examples illustrate this technique in both constant and variable velocity media. -Author

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.

Primaries (signal) and multiples (noise) often ex-hibit different kinematics and amplitudes (i.e., patterns) in time and space. Multidimensional prediction-error filters (PEFs) approximate these patterns to separate noise and signal in a least-squares sense. These filters are time-space variant to handle the nonstationarity of multioffset seismic data. PEFs for the primaries and multiples are estimated from pattern models. In an ideal case where accurate pattern models of both noise and signal exist, the pattern-based method recovers the pri-maries while preserving their amplitudes. In the more general case, the pattern model of the multiples is ob-tained by using the data as prediction operators. The pattern model of the primaries is obtained by convolv-ing the noise PEFs with the input data. In this situation, 3D PEFs are preferred to separate (in prestack data) the multiples properly and to preserve the primaries. Com-parisons of the proposed method with adaptive subtrac-tion with an 2 norm demonstrate that for a given multi-ple model, the pattern-based approach generally atten-uates the multiples and recovers the primaries better. In addition, tests on a 2D line from the Gulf of Mexico demonstrate that the proposed technique copes fairly well with modeling inadequacies present in the multiple prediction.

Wind a wire onto a cylinder to create a helix. I show that a filter on the 1-D space of the wire mimics a 2-D filter on the cylindrical surface. Thus 2-D convolution can be done with a 1-D convolution program. I show some examples of 2-D recursive filtering (also called 2-D deconvolution or 2-D polynomial division). In 2-D as in 1-D, the computational advantage of recursive filters is the speed with which they propagate information over long distances. We can estimate 2-D prediction-error filters (PEFs), that are assured of being stable for 2-D recursion. Such 2-D and 3-D recursions are general-purpose preconditioners that vastly speed the solution of a wide class of geophysical estimation problems. The helix transformation also enables us the partial-di#erential equation of wave extrapolation as though it was an ordinary-di#erential equation. INTRODUCTION This paper introduces and expounds a basic principle that promises wide-ranging practical applications: A multiple-dimensional Ca...

Suppressing artifacts in prestack reverse time migration

- R P Fletcher
- P J Fowler
- P Kitchenside

Fletcher, R. P., P. J. Fowler, and P. Kitchenside, 2005, Suppressing artifacts in
prestack reverse time migration: 75th Annual International Meeting, SEG,
Expanded Abstracts, 2049-2051.

2D deconvolution imaging condition for shot profile migration

- A Valenciano
- B Biondi

Valenciano, A., and B. Biondi, 2003, 2D deconvolution imaging condition
for shot profile migration: 73rd Annual International Meeting, SEG, Expanded Abstracts, 1059-1062.