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Reverse time migration in spatial frequency domain
Abstract
During the past decade, finite‐difference methods have become important tools for direct modeling of seismic data as well as for certain interpretation processes. One of the earliest applications of these methods to seismics is the pioneering contribution of Alterman who, in a series of papers (Alterman and Karal, 1968; Alterman and Aboudi, 1968; Alterman and Rotenberg, 1969; Alterman and Loewenthal, 1972) demonstrated the usefulness of such numerical computations for the propagation of seismic waves in elastic media. A clear exposition of these techniques, as well as a comparison of results obtained from them with the corresponding analytical solutions, can be found in Alterman and Karal (1968). This subject was further developed and extended to more complicated models by Boore (1970), Ottaviani (1971), and Kelly et al (1976). Claerbout introduced a somewhat different finite‐difference approach (Claerbout, 1970; Claerbout and Johnson, 1971) for modeling the acoustic waves which often dominate the reflection seismogram. In his approach, the original wave equation, which governs the propagation of the acoustic waves, is modified in such a way so as to allow the propagation of either only upcoming or only downgoing waves. By moving the coordinate frame with the downgoing waves, Claerbout showed that one could greatly reduce computation time. Using the same concepts, he showed (Claerbout and Doherty, 1972) how to use a similar scheme for migrating a seismic section by downward continuation of the upcoming waves. This migration method is an interesting extension of the ideas of Hagedoorn (1954) and was found to be extremely useful with real data (Larner and Hatton, 1976; Loewenthal et al, 1976).
... In exploration seismics several methods have been introduced to image complex structures including steeply dipping interfaces. A promising technique called reverse time migration (RTM) was originally introduced by Baysal et al. (1983), Loewenthal &Mufti (1983), andMcMechan (1983). However, due to the required computing power, it was not until after 2000 that RTM became a practicable method. ...
... In exploration seismics several methods have been introduced to image complex structures including steeply dipping interfaces. A promising technique called reverse time migration (RTM) was originally introduced by Baysal et al. (1983), Loewenthal &Mufti (1983), andMcMechan (1983). However, due to the required computing power, it was not until after 2000 that RTM became a practicable method. ...
The application of ultrasonic testing in civil engineering started several decades ago, mainly limited to pulse velocity measurements in transmission mode for concrete quality control. Starting in the late 1990s, more and more emphasis was put on echo techniques for structural investigations of concrete constructions. Starting with the arrival of point contact transducers without the need of coupling agents these techniques have raised the capabilities of nondestructive testing (NDT) to a new level. Ultrasonic testing (UT) in civil engineering so far mainly benefitted from methods developed for traditional NDT of metals.
However, there are still gaps in the currently applied technologies due to their inherent limitations. These gaps might be at least partially closed by applying geophysical techniques, which have been developed for a totally different scale in terms of penetration depths and wavelengths but aiming at similar physical/mathematical challenges (e.g. complex geometries, multiple scattering). This thesis deals with two examples of geophysical methods, which have been adapted to concrete testing.
One of the major drawbacks of the imaging techniques currently applied in ultrasonic echo testing for concrete is the limitation to simple geometries. SAFT (synthetic aperture focusing technique), which is widely and successfully used in many variants in NDT, is closely related to geophysical Kirchhoff or Stolt migration. All these techniques aim to focus the reflected energy recorded at the surface back to the reflector, but only direct single reflections are correctly dealt with. Vertical reflectors or the backside of internal features in a construction can`t be imaged. In this thesis, Reverse Time Migration (RTM), another geophysical imaging technique based on the correlation of forward and backward propagating wavefields, is proposed as an alternative for imaging complex geometries. It is shown, that vertical boundaries as those at thickness changes of foundations slabs can be imaged correctly as well as the full geometry of inclusions (voids, tendon ducts). Examples are given based on simulated and measured data. RTM has some drawbacks, as the significant computation time required and artifacts mainly close to the surface, which require additional research and development before widespread practical application in NDT.
The second development documented in this thesis is the adaption of algorithms borrowed from seismology to detect subtle changes in concrete due to various loads or degradation in ultrasonic transmission measurements. Since about 10 years several research groups are working on the application of Coda Wave Interferometry (CWI) to evaluate very small changes in the elastic wave velocity (e. g. by stress) or changes in the scattering pattern (e.g. by cracks) inside concrete. Various lab investigations, but also a few first field experiments have shown, that this technology is in fact able to contribute to monitoring of concrete constructions, revealing changes in the material properties (here e. g. by stress or temperature) and to localize the affected area. New instrumental developments as robust and reliable ultrasonic transducers help to path the way for long term implementation in concrete infrastructure (e.g. bridges). Methods to separate different influence factors or to simplify imaging are currently under development.
... Furthermore, the Kirchhoff integral method [2] can describe the propagation of waves in only smooth media, making it difficult to obtain an accurate imaging profile of complex structures. An alternative technique known as reverse time migration, a seismic data processing method based on extrapolating the two-way wave equation, can be used to migrate seismic reflection data to obtain subsurface images that effectively describe geological structures [3], [4]. This approach has many advantages, such as the ability to utilize various waveforms (e.g., reflected waves, multiples, and rotary waves) and the lack of a limit on the interface angle of inclination. ...
Reverse time migration based on the two-way wave equation using cross-correlation imaging can effectively yield migrated sections. However, when extrapolating the wavefield, reflected waves produce high-amplitude, low-frequency artefacts, so the imaging quality is a serious concern. Applying a Laplace filter to the migrated image can effectively suppress noise. However, results of applying this method are often polluted by discontinuous events and residual noise. To avoid these limitations, this paper proposes a Laplace filter by seek dip angle of underground structure. Based on an analysis of the mechanism which responsible for generating low-frequency noise, this paper analyses the basic principle and application of Laplace filter, and the low-frequency and high-amplitude characteristics of existing problems when extrapolating the wavefield. The low-frequency noise of the dipping strata and horizontal layer is different in imaging result, so this paper proposes establishing a Laplace operator based on dip angle of formation that are approximately normal to implement the filter. The results of experiments on models containing simple and complex structures show that the Laplace filter related to dip angle is effective and can better eliminate the low-frequency noise generated by back-reflected waves while maintaining the continuity of the events and protecting the effective signal.
... Seismic imaging has been used for decades to map impedance discontinuities in the subsurface from data acquired at the surface of the earth. In reverse time migration (RTM) (Baysal et al., 1983;Loewenthal and Mufti, 1983), one such seismic imaging method, the subsurface reflectivity is estimated with a three-step procedure. First, a forward wavefield mimicking the original acquisition is numerically modeled. ...
Extended least-squares inversion is superior to stack-based least-squares inversion for imaging the subsurface because it can better account for amplitude-versus-offset (AVO) effects as well as residual moveout (RMO) effects induced by erroneous velocity models. Surface-offset extensions have proved to be a robust alternative to angle gathers as well as subsurface extensions when applied to narrow-azimuth (NAZ) data acquisitions, especially when using erroneous velocity models. As such, least-squares reverse time migration (LSRTM) applied to surface-offset gathers (SOGs) obtains accurate surface-offset-dependent estimates of the reflectivity with better AVO behavior, while respecting curvatures of the events in the gathers. Nevertheless, the computational expense incurred by SOG demigration generally renders this process unfeasible in many practical situations. We exploit a compression scheme for SOGs that captures AVO and some RMO effects to improve efficiency of extended LSRTM. The decompression operator commutes with the demigration operator, so gathers compressed in the model domain may be decompressed in the data domain. This obviates the need to demigrate all SOGs, requiring only the demigration of a few compressed gathers. We demonstrate the accuracy of this compression, both in the model and data domains with a synthetic 2D data set. We then use our model-compression/data-decompression scheme to SOG-extended iterative LSRTM for two field data examples from offshore Brazil. These examples demonstrate that our compression can capture most AVO and some RMO information accurately, while greatly improving efficiency in many practical scenarios.
... Baysal et al. (1983) and Levin (1984) expound the principle and implementation method of RTM. Using the 2D acoustic wave equation, McMechan et al. (1983) achieve RTM in continuous and variable-speed media, and Loewenthal and Irshad (1983) modify the finite-difference (FD) method in the space domain and propose RTM based on the twoway wave equation. Wu et al. (1996) use high-order FD to perform RTM and discuss the advantages of high-over low-order FD. ...
Compared with one-way wave equation migration and ray-based migration, reverse time migration (RTM) using the two-way wave propagation information can produce accurate imaging result for complex structures. Its computational accuracy and efficiency are mainly determined by numerical method for wavefield simulation. When using traditional regular grids for seismic modeling, scattering artifacts may occur due to the stepped approximation of layer interfaces and rugged topography. On the other hand, the irregular grids requires complex grid generation algorithm, despite having certain geometric flexibility. Mesh-free RTM can effectively reduce the scattered noise under regular grids and avoid the extra computation in the process of irregular grids generation. For the implementation of mesh-free RTM method, an algorithm with fast generation of node distributions is used to discretize the underground velocity model, and radial-basis function generated finite-difference (RBF-FD) is used to realize the numerical simulation of wave propagation, cross-correlation imaging condition is adopted for imaging. The mesh-free RTM method which has both flexibility of simulation region and abundance of wavefield information, reduces the storage required for reverse time migration, shows the potential of high-accuracy migration in the case of undulating surface and provides more accurate migration imaging results for oil and gas exploration under complex geological conditions.
... For imaging the sources of scattered seismic wave arrivals in crustal and exploration studies, reverse time migration (RTM) was introduced in the 1980s (e.g., Baysal et al., 1983;Chang & McMechan, 1989;Loewenthal & Mufti, 1983;McMechan, 1983;Whitmore, 1983). It generally involves an imaging principle to extrapolate recorded seismic wavefields back to the locations from which they originated through scattering from buried heterogeneities (Claerbout, 1971). ...
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.
... Reverse time migration (RTM) aims to construct an image of subsurface reflectors by computing numerical solutions to wave equations. Proposed in the 1980s (Baysal et al., 1983;Loewenthal and Mufti, 1983;McMechan, 1983), the method relies on the calculation of two wavefields: a forward-propagated source wavefield and a backward-propagated receiver wavefield. The dot product of these two wavefields generates the desired image of the subsurface reflectors. ...
From a mathematical perspective, it is desirable to apply the adjoint operator for back propagating the receiver wavefield for migration of data residuals for inversion. For frequency-domain direct solvers, it is straightforward to apply the adjoint operator, whereas in most real applications of time-domain finite-difference (TDFD) stencils, the forward-propagation kernel function is reused for backward propagation for simplicity. However, when applying the exact adjoint operator, the migration result will be different, especially when strong variations exist in the velocity model. Actually, these three operators (forward, backward, and adjoint of the forward) can be expressed in matrix forms under the Born approximation for acoustic wave equations. These expressions linearly relate model perturbations to recorded seismic data. Every element in the matrices is well-defined, and all involved operations, such as the imaging condition and wavefield sampling, are included in the closed-form matrix expressions. These matrix expressions provide a platform for analyzing seismic modeling and inversion via mature linear algebra methodologies and provide clear strategies for developing computer algorithms. By analyzing the similarity of the matrix expressions, one can find that the time stepping approaches for all three operators are essentially the same. Based on this observation, a new time-marching stencil can be designed to realize the TDFD adjoint operator. Compared with traditional reverse time migration, the new method using the adjoint operator can provide better image quality, especially at sharp velocity boundaries.
... The reverse time migration (RTM) method was proposed by geophysicists, [1][2][3][4][5] and now it has been used for ultrasonic detection and imaging by bulk waves. [6][7][8][9][10] Wilcox et al. adopted the guided waves to accurately locate the defects in the rod by removing the effect of dispersion. ...
A modified reverse time migration (RTM) method for dispersed flexural waves is proposed to detect and image the crack in a thin plate. The key point of this method is to take the time reversal (TR) of the expanded flexural wave at a receiver as the backward wave in the cross-correlation imaging condition of RTM, thus the expanded backward wave will be re-compressed also due to dispersion of the flexural wave. The backward wave is time reversed again when arriving at a certain point on the plate, and then correlated to the forward wave derived from the original source. The image of the crack is obtained by superimposing the cross-correlation results of many source-receiver pairs in transmitting and receiving arrays. The experimental studies on imaging a crack in a thin plate are conducted, and the transmitted and received waveforms are recorded with high fidelity by a laser vibrometer, thus, ensuring that an accurate and clear image of the crack is obtained by this modified RTM process.
... Prestack reverse time migration (RTM) was initially proposed by Whitmore (1983) at the SEG annual meeting. Following this study, Baysal et al., (1983), Loewenthal and Mufti (1983) and McMechan (1983) applied the RTM algorithm to poststack P-wave seismic data. RTM, which exploits the two-way wave equation for wavefield propagation, is theoretically capable of mapping various wavefields (e.g. ...
Compared with other imaging algorithms (e.g., ray-based, one-way wave equation), reverse time migration (RTM) based on the two-way wave equation exhibits greater superiority, especially in handling steeply dipping structures. However, imaging with conventional single-component seismic data is unsuited for some complicated structures (e.g., gas clouds). Elastic RTM, which is based on the elastodynamic equation and uses multi-component seismic data to extract PP and PS reflectivity and subsurface information, can more consistently reproduce the characteristics of elastic wave propagation in real Earth media, resulting in seismic images that more accurately characterize the subsurface. To begin with, we exploit the first order stress-velocity equations to extrapolate the elastic vector wavefield, then the P- and S-wavefields are separated by computing the divergence and curl operator of the extrapolated particle-velocity wavefield. Then, imaging profiles with pure wave modes are computed by applying the source normalized cross-correlation imaging condition, thus avoiding crosstalk between unseparated wave modes. To address the polarity reversal problem of the converted image, we propose an alternative method in the common-shot domain. We also develop an efficient method that reconstructs the source wavefield in the reverse time direction to save storage in the GPU and to avoid large input/output in the elastic reverse time migration. During the forward modeling, the method only saves the particle-velocity wavefield of all time intervals within an efficient absorbing boundary and the total wavefields in the final time interval. When we extrapolate the receiver wavefield in the reverse time direction, we simultaneously reconstruct the total source wavefields via the saved wavefields. Numerical examples performed with the graben and Marmousi2 models have shown that the polarity reversal correction method works, and elastic reverse time migration can accurately characterize complicated structures.
... These methods entail solving costly simulations via finite-difference techniques (Claerbout, 1970;Ristow and Rühl, 1994). These types of migration algorithms are called reverse time migration or two-way wave-equation migration (Baysal et al., 1983;Loewenthal and Mufti, 1983;McMechan, 1983;Whitmore, 1983;Levin, 1984). ...
We have investigated the problem of designing the forward operator and its exact adjoint for two-way wave-equation least-squares migration. We study the problem in the time domain and pay particular attention to the individual operators that are required by the algorithm. We derive our algorithm using the language of linear algebra and establish a simple path to design forward and adjoint operators that pass the dot-product test. We also found that the exact adjoint operator is not equal to the classic reverse timemigration algorithm. For instance, one must pay particular attention to boundary conditions to compute the exact adjoint that accurately passes the dot-product test. Forward and adjoint operators are adopted to solve the so-called least-squares reverse time migration problem via the method of conjugate gradients.We also examine a preconditioning strategy to invert extended images.
... Prestack reverse time migration (RTM) was initially proposed by Whitmore (1983) at the SEG annual meeting. Following this study, Baysal et al., (1983), Loewenthal and Mufti (1983) and McMechan (1983) applied the RTM algorithm to poststack P-wave seismic data. RTM, which exploits the two-way wave equation for wavefield propagation, is theoretically capable of mapping various wavefields (e.g. ...
Compared with other imaging algorithms (e.g., ray-based and one-way wave equation), reverse time migration (RTM) based on two-way wave equation exhibits great superiority, especially in dealing with steeply dipping structures. However, imaging with conventional single-component seismic data may become imperfect in some complicated structures (e.g., gas clouds). The elastic reverse time migration based on the elastodynamic equation using multi-component seismic data can extract PP and PS reflectivity containing subsurface information, thus it can be more consistent with characteristics of elastic wave propagation in the real earth's medium, and resulting seismic images can more accurately characterize the subsurface. To begin with, we employ the first-order velocity-stress equations to implement extrapolation of elastic vector wavefields, and separate P- and S-wavefields by computing the divergence and curl operators of the extrapolated velocity vector wavefields. Then, imaging data with pure wave modes can be computed by applying the source normalized cross-correlation imaging condition, thus avoiding the crosstalk between the unseparated wave modes. To address the polarity reversal problem of converted images, we propose an alternative method in the shot domain. We also develop an efficient method that reconstructs source wavefields in the reverse time direction to save storage in CPU and avoid large input/output for elastic reverse time migration. During the forward modeling, the method only saves velocity vector wavefields of all time interval within an efficient absorbing boundary and total wavefields in the last time interval. When we extrapolate the receiver wavefields in reverse time direction, simultaneously, we reconstruct the total source wavefields by the saved wavefields. Numerical examples with a graben and Marmousi2 models show that the polarity reversal correction method works well and that elastic reverse time migration can generate accurate images for complicated structures.
... These methods entail solving costly simulations via finite-difference techniques (Claerbout, 1970;Ristow and Rühl, 1994). These types of migration algorithms are called reverse time migration or two-way wave-equation migration (Baysal et al., 1983;Loewenthal and Mufti, 1983;McMechan, 1983;Whitmore, 1983;Levin, 1984). ...
The selection of imaging conditions is one of the most critical factors determining the quality of reverse time migration (RTM) images. Among the widely used imaging conditions, the cross-correlation imaging condition (CCIC) consistently delivers high-resolution images. However, it is accompanied by substantial calculational costs and I/O tasks, particularly in 3D scenarios. In contrast, the excitation amplitude imaging condition (EAIC) offers advantages in computational efficiency, low storage requirements, and high precision. Nevertheless, it suffers from image distortion when dealing with multi-path propagation or strong reflection interfaces. The local Nyquist cross-correlation imaging condition (LNCIC) effectively combines the advantages of the two aforementioned imaging conditions. It uses the local wavefield near the time corresponding to the maximum amplitude at each grid point for imaging, and introduces the Nyquist sampling theorem to establish the search time step. This approach offers the benefit of high imaging quality while maintaining low storage cost. In this paper, we adopt an adaptive finite difference operator to solve the eikonal equation and calculate the accurate first-arrival traveltimes, thereby modify LNCIC and further enhancing the imaging accuracy. The effectiveness of the proposed method is demonstrated through numerical examples, including the Marmousi model, noise-resistance tests, and field data applications.
Reverse time migration (RTM) can provide high-quality seismic images and is one of the most advanced migration methods. Imaging condition method is a crucial component in RTM. Different imaging conditions show different abilities to obtain image amplitude, physical validity, and resolution. The cross-correlation imaging condition can achieve high imaging accuracy for complex structures. Nevertheless, this approach necessitates the storage of almost the entire source wavefields, which demands a substantial amount of storage space and consequently reduces the computational efficiency of reverse time migration. In contrast, the excitation amplitude imaging condition, which relies solely on wavefield information at the imaging time, eliminates the necessity for extensive storage, thereby enhancing the computational efficiency of reverse time migration. However, conventional methods determine the imaging time by identifying the maximum amplitude of the source wavefield at each grid point. In scenarios involving large offsets or strong reflection interfaces, this approach can lead to inaccuracies in determining the imaging time. In this paper, we utilize an advanced adaptive finite-difference operator method to solve the eikonal equation for determining imaging time. This approach markedly enhances the accuracy of first-arrival traveltimes calculations at near offsets, which is especially critical for seismic imaging. Numerical examples from simple model, the Marmousi model, and field seismic data demonstrate the effectiveness of our proposed method.
Multi-parameter elastic full waveform inversion (EFWI) provides a more realistic depiction of the subsurface models than the standard acoustic approximation. In practice, however, the significant additional cost and interdependency between the unknown parameters (cross-talks) hinder the application of such algorithms. Diffusion model-based regularization can be used to improve the inversion results while simultaneously injecting prior information into the solution. The main challenge here is how to inject such priors into the EFWI iterations that can better complement the solution’s evolution. To address this challenge, we incorporate a model wavenumber continuation process into a diffusion model-based regularization contribution to multi-parameter EFWI. To do so, we promote a sampling strategy such that at the early iteration, the proposed regularization updates account for the low wavenumber component more and increase progressively with the iteration. We first train the diffusion model on elastic moduli images in an unsupervised manner and incorporate the trained model during the EFWI inversion. We deliberately use single-component measurements, which is the most common acquisition scenario, during the inversion to demonstrate the effectiveness of our regularization. At the inference stage, the proposed framework provides more accurate solutions with negligible additional computational cost compared to several conventional regularization algorithms.
Improving the imaging accuracy of geological formations under dual complex conditions (including complex surface and structure) is essential for precisely illustrating structural morphology and understanding reservoir characteristics. The multi-focus imaging is a real surface imaging method that takes into account signal-to-noise ratio and resolution. Drawing upon the principles of paraxial ray theory and Hubra's two wavefronts theory, this approach employs a global optimization inversion algorithm to determine the radii and exit angles of the two wavefronts. Furthermore, it incorporates a non-hyperbolic travel time formula for accurate correction. By combining receiving channels from different CMP channels within the same Fresnel band radius, this method effectively enhances both signal-to-noise ratio and resolution of seismic data. The multi-focus imaging technique is a surface imaging method that considers both signal-to-noise ratio and resolution. Drawing upon the principles of paraxial ray theory and Hubra's two wavefronts theory, this approach employs a global optimization inversion algorithm to determine the radii and exit angles of the two wavefronts. Furthermore, it incorporates a non-hyperbolic travel time formula for accurate correction. By combining receiving channels from different CMP channels within the same Fresnel band radius, this method effectively enhances both signal-to-noise ratio and resolution of seismic data.
Reverse-time migration (RTM) is widely regarded as one of the most accurate migration methods available today. A crucial step in RTM involves extending seismic wavefields forward and backward. Compared to the conventional central finite difference (CFD) scheme, the combined compact difference (CCD) scheme offers several advantages, including a shorter difference operator and the suppression of numerical dispersion under coarse grids. These attributes conserve memory and enhance effectiveness while maintaining the same level of differential precision. In this article, we begin with the five-point eighth-order CCD scheme and utilize the least squares method and Lagrange multiplier method to optimize the difference coefficients. This optimization is guided by the concept of dispersion-relation-preserving (DRP). The result is the acquisition of an optimized combined compact difference (OCCD) scheme, further enhancing the ability to suppress numerical dispersion. We thoroughly compare and analyze dispersion relationships and stability conditions. In addition, we examine several crucial steps in the RTM of the second-order acoustic wave equation. These steps include absorption boundary conditions, boundary storage strategy, and Poynting vector imaging conditions. Finally, we apply both the CCD and OCCD schemes in the RTM of the layered model, graben model, and SEG/EAGE salt model. We compare these results with those obtained from CFD's RTM. Numerical findings demonstrate that, in contrast to the CFD scheme, the CCD scheme effectively suppresses numerical dispersion and enhances imaging accuracy. Moreover, the optimized OCCD scheme further improves the ability to suppress numerical dispersion and can obtain better imaging results, which is an effective reverse time migration method suitable for coarse grid conditions.
Seismic data is the primary way to study the subsurface structure and properties. A conventional seismic sensor like geophone or accelerometer measures particle velocity or acceleration at a local point only, while Distributed Acoustic Sensing (DAS) measures dynamic strain along the fiber optic cable at densely spaced sample points where strain rate is obtained over certain gauge length interval. Therefore, DAS measures subsurface properties with high sampling resolution and large coverage. When an optical fiber is installed in a well, DAS can provide continuous, dense downhole recording. However, currently most of the seismic processing, imaging and inversion techniques are developed for geophone data. These well-established techniques can be readily and properly utilized if DAS data is transformed into geophone measurements such as particle velocity. In this study, we present a recurrent neural network framework to perform this transformation. This effectiveness of the deep learning-based mapping is then demonstrated with a field measurement data, showing that DAS data can be transformed into particle velocity accurately and robustly using the proposed deep-learning approach.
The reflectivity of the subsurface can be precisely determined using least-squares reverse time migration (LSRTM). Since LSRTM necessitates solving the wave equation, the numerical solution method of the wavefield directly determines the quality of the migration image. The conventional LSRTM method usually uses the finite difference method based on a regular grid to calculate the wavefield. Due to the stepwise approximation of an irregular surface with a regular grid, scattering noise may occur in the propagation of the wavefield, which affects the quality of the image. In addition, the conventional LSRTM cannot effectively handle the models with rugged topography. The finite difference method generated by radial basis functions (FD-RBF) is a mesh-free method and can construct interpolation functions to solve the wave equations numerically according to arbitrarily distributed spatial coordinate points. Therefore, we use the FD-RBF method to develop a mesh-free LSRTM approach to eliminate the influence of the inherent limitation of a regular grid on the imaging. Numerical examples show that the mesh-free LSRTM method can better represent the curved or steep interface within the model and is also suitable for models with rugged topography. The LSRTM method can provide higher-quality images and effectively reduce the memory required for calculations.
The excitation amplitude imaging condition (EAIC) is a high-resolution, computationally efficient, and low-storage imaging condition in reverse time migration (RTM). However, when there are strong reflection interfaces in the velocity model, they will produce low-frequency artifacts, which seriously contaminate the RTM image. The artifacts can be removed by the wavefield decomposition algorithm, but this process always performed by analytic time wavefield extrapolation, which needs extra wavefield extrapolation. Furthermore, an extra source wavefield extrapolation is required to determine the excitation time before the migration. Thus, the additional wavefield extrapolations can seriously damage the computationally efficient advantage of the EAIC. By taking advantage of the directivity and low storage of excitation amplitude, we present a low-frequency artifact suppression method with no extra wavefield extrapolation. Poynting vector, reference traveltime and minimum amplitude threshold are combined to constraint the excitation amplitude updating process, and it makes the excitation amplitude more consistent with the definition of excitation criterion. We can directly obtain a noise-free excitation amplitude without the source wavefield decomposition. Instead of the analytic time wavefield extrapolation, the time-bin technique and the windowed Hilbert transform are combined to achieve the receiver wavefield decomposition only at the excitation time. The numerical results show that our method can effectively suppress the low-frequency artifacts in the image with no extra wavefield extrapolation.
The problems of large computation, large storage and low-frequency noises have been the key factors limiting the development of elastic wave pre-stack reverse-time migration (RTM). Most research has centered on the wave equation of first-order velocity stress and using Helmholtz separation to get pure primary and secondary waves. However, the cost is enormous and the amplitude and phase of wavefields will be changed. So, we use the second-order P- and S-wave decoupling equations to construct seismic wavefields that reduce the number of variables, and the storage space required for boundary wavefields is reduced by ∼78%. Then we bring the boundary-saving approach into our study during the wavefield extrapolation process so that we can reconstruct the source wavefields completely. Finally, for the wave equations of second-order displacement used in this study, the Poynting vector expressed by the traditional method was not applicable. Therefore, we derived the formula of the energy flux density vector represented by displacements and used it in directional traveling wave decomposition to suppress low-frequency noise in the imaging profile. Subsequently, based on the inner product imaging conditions, the RTM of the 2-D salt dome model was calculated. The results illustrate the effectiveness and practicability of the proposed solution.
The Rakhine Basin in the northeastern Bay of Bengal is an active field in hydrocarbon exploration and development. It contains fault structures and steeply sloping stratigraphic reservoirs, both primary features of interest for hydrocarbon exploration that needs to be accurately imaged to improve the interpretation of seismic data and facilitate the accurate identification of features of interest. Although faults are an indicator of possible hydrocarbon traps, they are difficult to identify in seismic images using traditional stack or prestack time migration due to the rather complex behaviors of wave propagation. On the other hand, prestack depth migration (PSDM) can significantly improve the accuracy of seismic images, especially of complex subsurface structures such as faults, folds, overthrusts, and salt domes. Among the various PSDM approaches, reverse time migration (RTM) has been shown to be the most powerful. Here, we show how PSDM-RTM can significantly improve the representation of fault structures and steeply dipping structures in seismic images from field data collected in Rakhine Basin, which is characterized by complex geology including stratigraphic and strati-structural traps as well as complex channel systems. Typically, these structures appear heavily blurred and are difficult to identify using normal stack and prestack time migration. We demonstrate that they become clearer and easier to detect with the PSDM–RTM approach, making this approach particularly suitable for seismic interpretations of geologically complex areas within the context of hydrocarbon prospecting.
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.
Prismatic waves carry steeply dipping structural information that primaries cannot contain. Therefore, prismatic waves are separately used in some migration methods to improve the illumination and imaging effect on steeply dipping structures. Least-squares reverse time migration of prismatic waves (LSRTM-P) can produce high-resolution images with improved steeply dipping structures. However, viscoelasticity exists widely on the Earth, which poses great difficulty for imaging. The effect of attenuation on prismatic waves is difficult to be compensated when conducting LSRTM-P because prismatic waves have three propagation paths. To overcome this problem, a
Q
-compensated LSRTM (
Q
-LSRTM)-P method is proposed by deriving
Q
-compensated forward-propagated operators and backward-propagated adjoint operators of prismatic waves, which compensates for
Q
attenuation along all the three propagation paths of prismatic waves. The proposed
Q
-LSRTM-P is conducted to update the image after applying the conventional
Q
-LSRTM. Besides, the proposed method can be adapted to the irregular surface media. Numerical examples on two synthetic and a field datasets verify that our method can produce better imaging results with clearer steeply dipping structures, higher signal-to-noise ratio (SNR), higher resolution, and more balanced amplitude than noncompensated LSRTM-P and conventional
Q
-LSRTM.
Frequency-domain finite-difference (FDFD) modeling plays an important role in exploration seismology. However, a major disadvantage of FDFD modeling is the computational cost, especially for large-scale models. By compactly distributing nonzero strips, the elongated stencil helps to generate a narrow-bandwidth impedance matrix, improving computational efficiency without sacrificing numerical accuracy. To further improve the accuracy and efficiency of modeling, we have developed an optimal FDFD method with an elongated stencil for 2D acoustic-wave modeling. The Laplacian term is approximated using the directional-derivative method and the average-derivative method. The dispersion analysis indicates that this elongated-stencil-based method (ESM) achieves higher accuracy than other finite-difference methods with the elongated stencil, and it is more suitable for large grid-spacing ratios. To keep the phase-velocity error within 1%, 15-point and 21-point schemes in the ESM only require approximately 2.28 and 2.19 grid points per wavelength, respectively, when the grid-spacing ratio, namely, the ratio of directional sampling intervals, is not less than 1.5. Moreover, we also adopt a variable-stencil-length scheme, in which the stencil length varies with the velocity, to further reduce the computational cost in frequency-domain modeling. Several numerical examples are presented to demonstrate the effectiveness of our ESM.
3D Reverse-time migration (RTM) is a powerful technique for imaging complex geologic structures. This approach requires a significant computational effort, demanding a high amount of memory for storing the source's wavefields, consequently leading to a high cost to perform the imaging condition. Thus, this work aims to reduce these problems by introducing a dynamic approach (DA) that considers the sparsity of the wavefield in the first periods of propagation. The RTM combined with the DA (RTM-DA) approximates the computational domain to the propagation domain, which is the region delimited by the wavefront. In practical terms, the computational domain expands together with the wavefront, reflecting in a very significant economy of memory and reduction of processing time when compared to the conventional RTM, which we denominate as a static approach (RTM-SA). To reduce the sparsity of the wavefields in the first periods of propagation, we have built an empirical 3D filter that maps each timestep of the wavefield and gives the coordinates to approximate the computational domain to the propagation domain. We compare both approaches using the 3D SEG/EAGE Salt model and demonstrate that the RTM-DA is more efficient than the RTM-SA in terms of memory consumption and computational time, preserving the quality of the seismic image.
We consider the problem of inhomogeneous subsurface imaging using beam waves. The formulation is based on the ultra-wide-band phase-space beam summation (UWB-PS-BS) method, which is structured upon windowed Fourier transform (WFT) expansions of surface fields and sources. In this approach, the radiated field is given as a superposition of beam propagators. Here, we use the beams first for expanding the surface sources and the scattered data, and then for imaging where we use the backpropagation and cross-correlation of beams. This formulation enables a target oriented imaging approach, where we take into account only pairs of source and receiver beams that pass near a region of interest, and thus extract only the relevant data arriving from this region. It also leads to a priori sparse representation of both the beam domain data and the beam propagators. A physical cogent for the beam domain data is obtained under the Born approximation. The beam domain data can be approximated as the local interaction between the beam propagators and the medium reflectivity. Thus, one may interpret the beam domain data as a local Snell’s law reflection in the direction defined by the vector summation of the incident beam and backpropagated beam ray parameters. We demonstrate a physical model for the beam domain data and the salient features of the proposed imaging algorithm using numerical examples.
To investigate wavefield depth extrapolation using the full wave equation, we derive a new depth extrapolation scheme for migration using functions of the vertical wavenumber. We present a complete matrix multiplication formulation and approach to calculate the related mathematical functions of the vertical wavenumber and perform the depth extrapolation using matrix multiplication only. Because our proposed depth extrapolation algorithm involves only matrix multiplication, it is naturally applicable to parallel computations. Impulse response experiments demonstrate that our proposed migration method can achieve the same accuracy as full wave-equation migration using the finite-difference method, in terms of phase information, even for media with strong lateral velocity changes. In numerical experiments using a smoothed version of the 2D SEG/EAGE salt model, our proposed migration method provides an equivalent imaging result compared with RTM and a more accurate imaging result than migration using one-way propagators. Our proposed method has certain potential advantages over RTM using the same full wave-equation with fewer internal multiple scatterings and less data storage requirements. Our proposed method is a stable depth extrapolation scheme because the evanescent waves are well suppressed. The numerical experimental results on the synthetic model demonstrate the importance of suppressing evanescent waves in a full wave-equation-based depth-extrapolation scheme and migration for both imaging quality and computation cost.
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.
The safety and efficiency of tunnel construction depend on the knowledge of complex geological conditions. Hence, an accurate forward-prospecting technique is required to detect unexpected inhomogeneous geological structures ahead of tunnel construction. As an accurate method to image geological heterogeneities, elastic reverse-time migration (ERTM) is introduced to the field of tunnel forward-prospecting. However, in the tunnel environment, 1 ERTM images may suffer from the interference of crosstalk artifacts, which are caused by converted waves on tunnel surfaces. Therefore, considering the actual influence of the tunnel body, the decoupled non-conversion elastic equation was incorporated into the traditional ERTM. This method prevents the generation of converted waves but ensures indepedent P-and S-wave propagation. In addition, wave-mode separation for raw seismic data is required in the proposed approach. Synthetic examples based on the real geological environment of tunnel show that our method produces satisfactory results for both P-wave and S-wave imaging, and S-wave can produce a better imaging effect in tunnels. Finally, we applied our method to the seismic data obtained from a real highway tunnel construction site to demonstrate its performance in real-world applications. The results show that the migrated images can help accurately constrain the geologic formations ahead the tunnel face.
Cross-correlation imaging condition of reverse time migration generates high-amplitude artifacts in the reconstructed images. In addition, due to geometrical spreading and scattering attenuations, this imaging condition assigns amplitudes to the points of the reconstructed image that are not a true representative of the reflection coefficient of the scanned medium at those points. These all can lead to a reduction in quality and misinterpretation of the reconstructed images. In this paper, we proposed new imaging conditions to mitigate high-amplitude artifacts and to assign more accurate amplitudes to an RTM image by considering geometrical spreading and scattering attenuation in concrete members when horizontal shear waves are used for imaging. We used data obtained by transmitting horizontal shear waves to 3D synthetic homogeneous and concrete specimens and demonstrated the effectiveness of the new imaging conditions.
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.
A key technical challenge facing borehole acoustic reflection imaging is the uncertainty concerning the azimuth (i.e. the direction about the axis of the borehole) from which a reflected event arrives. In generating images of formation boundaries near the borehole this is referred to as azimuthal ambiguity. The phenomenon is comparable to the 3D out-of-plane effects encountered in imaging of 2D surface data. Systems with multiple receivers (e.g., the sonic scanner tool developed by Schlumberger) acquire sufficient information to uniquely image reflected wave energy at its correct azimuth. However, a full use of this information requires the wave data to be analyzed in the context of 3D, two-way elastic wave propagation and a suitable anisotropic medium. Here we summarize the development of a 3D anisotropic reverse time migration imaging scheme to sonic data, and demonstrate that within this imaging formulation azimuthal ambiguity is resolved for data from both monopole and dipole sources.
Conventional reverse time migration (RTM) may not produce high-quality images in areas with attenuation and severe topography because severe topographic surfaces have a great impact on seismic wave simulation, resulting in strong scattering and diffraction waves, and anelastic properties of the earth affect the kinematics and dynamics of seismic wave propagation. To overcome these problems, we have developed a Q-compensated topographic RTM method. In this method, a new viscoacoustic quasidifferential equation is introduced to simulate forward- and backward-propagated wavefields. The viscoacoustic equation has a lossy term and a dispersion term without memory variables, and it is solved by a hybrid spatial partial derivative scheme. A new stabilization operator is derived and substituted into the Q-compensated viscoacoustic quasidifferential equation to suppress high-frequency noise during the attenuated wavefield compensation. Numerical tests on a sag attenuating topographic model and an attenuating topographic Marmousi2 model demonstrate that our Q-compensated topographic RTM can produce accurate and high-quality images by correcting the anelastic amplitude loss and phase-dispersion effects. Finally, our method is tested on a field data set.
A three-dimensional high-order reverse-time migration (3-D HO-RTM) method is proposed to perform subsurface electromagnetic imaging with an ultra-wideband radar (UWBR) system consisting of a multi-input and multi-output antenna array. By using a UWBR system to collect temporal scattering signals, subsurface targets can be detected, and the image of targets can be obtained by imaging methods such as the back-propagation method, frequency-wavenumber migration technique, time-reversal mirror, and reverse-time migration method. The proposed HO-RTM method is based on the high-order finite-difference time-domain (HO-FDTD) method to significantly reduce the computational cost in the conventional RTM method. The measured data from an experimental lunar exploration system Chang'E-5 have been collected on a 7 m × 2.5 m × 2.5 m laboratory model with volcanic ash and validated by the 3-D HO-RTM method. Results show that all buried objects can be effectively identified by the HO-RTM, and its computer memory and CPU time are only 3.87% and 0.7128% of the conventional RTM method, respectively.
Backprojection imaging has recently become a practical method for local earthquake detection and location due to the deployment of densely sampled, continuously recorded, local seismograph arrays. While backprojection sometimes utilizes the full seismic waveform, the waveforms are often pre-processed and simplified to overcome imaging challenges. Real data issues include aliased station spacing, inadequate array aperture, inaccurate velocity model, low signal-to-noise ratio, large noise bursts and varying waveform polarity. We compare the performance of backprojection with four previously used data pre-processing methods: raw waveform, envelope, short-termaveraging/long-termaveraging and kurtosis. Our primary goal is to detect and locate events smaller than noise by stacking prior to detection to improve the signal-to-noise ratio. The objective is to identify an optimized strategy for automated imaging that is robust in the presence of real-data issues, has the lowest signal-to-noise thresholds for detection and for location, has the best spatial resolution of the source images, preserves magnitude, and considers computational cost. Imaging method performance is assessed using a real aftershock data set recorded by the dense AIDA array following the 2011 Virginia earthquake. Our comparisons show that raw-waveform backprojection provides the best spatial resolution, preserves magnitude and boosts signal to detect events smaller than noise, but is most sensitive to velocity error, polarity error and noise bursts. On the other hand, the other methods avoid polarity error and reduce sensitivity to velocity error, but sacrifice spatial resolution and cannot effectively reduce noise by stacking. Of these, only kurtosis is insensitive to large noise bursts while being as efficient as the raw-waveformmethod to lower the detection threshold; however, it does not preserve the magnitude information. For automatic detection and location of events in a large data set, we therefore recommend backprojecting kurtosis waveforms, followed by a second pass on the detected events using noise-filtered raw waveforms to achieve the best of all criteria.
The discontinuous-grid method can greatly reduce the storage requirement and computational cost in finite-difference modeling, especially for models with large velocity contrasts. However, this technique is mostly applied to time-domain methods. We have developed a discontinuous-grid finite-difference scheme for frequency-domain 2D scalar wave modeling. Special frequency-domain finite-difference stencils are designed in the fine-coarse grid transition zone. The coarse-to-fine-grid spacing ratio is restricted to 2ⁿ, where n is a positive integer. Optimization equations are formulated to obtain expansion coefficients for irregular stencils in the transition zone. The proposed method works well when teamed with commonly used 9- and 25-point schemes. Compared with the conventional frequency-domain finite-difference method, the proposed discontinuous-grid method can largely reduce the size of the impedance matrix and number of nonzero elements. Numerical experiments demonstrated that the proposed discontinuous-grid scheme can significantly reduce memory and computational costs, while still yielding almost identical results compared with those from conventional uniform-grid simulations. When tested for a very long elapsed time, the frequency-domain discontinuous-grid method does not show instability problems as its time-domain counterpart usually does.
The emergence of ultrasonic dry point contact (DPC) transducers that emit horizontal shear waves has enabled efficient collection of high-quality data in the context of a nondestructive evaluation of concrete structures. This offers an opportunity to improve the quality of evaluation by adapting advanced imaging techniques. Reverse time migration (RTM) is a simulation-based reconstruction technique that offers advantages over conventional methods, such as the synthetic aperture focusing technique. RTM is capable of imaging boundaries and interfaces with steep slopes and the bottom boundaries of inclusions and defects. However, this imaging technique requires a massive amount of memory and its computation cost is high. In this study, both bottlenecks of the RTM are resolved when shear transducers are used for data acquisition. An analytical approach was developed to obtain the source and receiver wavefields needed for imaging using reverse time migration. It is shown that the proposed analytical approach not only eliminates the high memory demand, but also drastically reduces the computation time from days to minutes.
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.
Imaging is a key step in seismic data processing. To date, a myriad of advanced pre-stack depth migration approaches have been developed; however, reverse time migration (RTM) is still considered as the high-end imaging algorithm. The main limitations associated with the performance cost of reverse time migration are the intensive computation of the forward and backward simulations, time consumption, and memory allocation related to imaging condition. Based on the reduced order modeling, we proposed an algorithm, which can be adapted to all the aforementioned factors. Our proposed method benefit from Krylov subspaces method to compute certain mode shapes of the velocity model computed by as an orthogonal base of reduced order modeling. Reverse time migration by reduced order modeling is helpful concerning the highly parallel computation and strongly reduces the memory requirement of reverse time migration. The synthetic model results showed that suggested method can decrease the computational costs of reverse time migration by several orders of magnitudes, compared with reverse time migration by finite element method.
Angle-domain common-image gathers (ADCIGs) transformed from the shotdomain common-offset gathers are input to migration velocity analysis (MVA) and prestack inversion. ADCIGs are non-illusion prestack inversion gathers, and thus, accurate. We studied the extraction of elastic-wave ADCIGs based on amplitude-preserving elastic-wave reversetime migration for calculating the incidence angle of P-and S-waves at each image point and for different source locations. The P-and S-waves share the same incident angle, namely the incident angle of the source P-waves. The angle of incidence of the source P-wavefield was the difference between the source P-wave propagation angle and the reflector dips. The propagation angle of the source P-waves was obtained from the polarization vector of the decomposed P-waves. The reflectors’ normal direction angle was obtained using the complex wavenumber of the stacked reverse-time migration (RTM) images. The ADCIGs of P-and S-waves were obtained by rearranging the common-shot migration gathers based on the incident angle. We used a horizontally layered model, the graben medium model, and part of the Marmousi-II elastic model and field data to test the proposed algorithm. The results suggested that the proposed method can efficiently extract the P-and S-wave ADCIGs of the elastic-wave reverse-time migration, the P-and S-wave incident angle, and the angle-gather amplitude fidelity, and improve the MVA and prestack inversion.
Least-squares migration (LSM) is commonly regarded as an amplitude-preserving or true amplitude migration algorithm that, compared with conventional migration, can provide migrated images with reduced migration artifacts, balanced amplitudes, and enhanced spatial resolution. Most applications of LSM are based on the constant-density assumption, which is not the case in the real earth. Consequently, the amplitude performance of LSM is not appropriate. To partially remedy this problem, we have developed a least-squares reverse time migration (LSRTM) scheme suitable for density variations in the acoustic approximation. An improved scattering-integral approach is adopted for implementation of LSRTM in the frequency domain. LSRTM images associated with velocity and density perturbations are simultaneously used to generate the simulated data, which better matches the recorded data in amplitudes. Summation of these two images provides a reflectivity model related to impedance perturbation that is in better accordance with the true one, than are the velocity and density images separately. Numerical examples based on a two-layer model and a small part of the Sigsbee2A model verify the effectiveness of our method.
Azimuthal ambiguity remains a key issue for borehole acoustic reflection imaging. This imaging issue, which occurs not only in borehole reflection imaging but also affects seismic when out of plane effects are not accounted for, is due to processing that treats recorded 3D waves as though they were 2D. As a solution, the 4-C dipole acoustic well logging technique can be applied; this approach resolves the azimuthal ambiguity problem by incorporating the directional information contained in the recorded shear wave signals. A migration procedure can then be carried out on the result to determine an image with structures unambiguously placed. In this paper, a 3D reverse time migration in the borehole environment is proposed for this purpose, and is evaluated using a simulated data set. Directional information for correct imaging of the structures outside the borehole appear to be stably obtained.
Presentation Date: Wednesday, October 19, 2016
Start Time: 3:10:00 PM
Location: 168
Presentation Type: ORAL
We have developed a method for accelerating the convergenceof iterative least-squares migration. The algorithmuses a pseudodifferential scaling (dip and spatially varyingfilter) preconditioner together with a variant of conjugategradient (CG) iteration with iterate-dependent (flexible) preconditioning.The migration is formulated without the imagestack, thus producing a shot-dependent image volume thatretains offset information useful for velocity updating andamplitude variation with offset analysis. Numerical experimentsindicate that flexible preconditioning with pseudodifferentialscaling not only attains considerably smaller datamisfit and gradient error for a given computational effort,but also produces higher resolution image volumes withmore balanced amplitude and fewer artifacts than is achievedwith a nonpreconditioned CG method. © 2016 Society of Exploration Geophysicists. All rights reserved.
The motion of a heterogeneous fluid sphere due to an impulsive point source is calculated by several finite difference schemes and is interpreted in terms of reflected and diffracted pulses. The deviations due to heterogeneity from the results for the homogeneous sphere are calculated explicitly. Calculations show the change in amplitude of pulses and dispersion due to continuous variation of compressional velocity c, as compared with discontinuities in c. © 1968, The Seismological Society of Japan, The Volcanological Society of Japan, The Geodetic Society of Japan. All rights reserved.
Since the first volume of this work came out in Germany in 1924, this book, together with its second volume, has remained standard in the field. Courant and Hilbert's treatment restores the historically deep connections between physical intuition and mathematical development, providing the reader with a unified approach to mathematical physics. The present volume represents Richard Courant's second and final revision of 1953. © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. All rights reserved.
The multidimensional scalar wave equation at a single frequency is split into two equations. One controls the downgoing transmitted wave; the other controls the upcoming reflected wave. The equations are coupled, but in many reflection seismology situations the transmitted wave may be calculated without consideration of the reflected wave. The reflected wave is then calculated from the transmitted wave and the assumed velocity field. The waves are described by a modulation on up‐ or downgoing plane waves. This modulation function is calculated by difference equations on a grid. Despite complicated velocity models (steep faults, buried focus, etc.), the grid may be quite coarse if waves of interest do not propagate at large angles from the vertical. A one‐dimensional grid may be used for a two‐dimensional velocity model. With approximations, a point source emitting waves spreading in three dimensions may be included on the one‐dimensional grid. Calculation time for representative models is a few seconds. Phenomena displayed are interference, spherical spreading, propagation through focus, refraction, and diffraction. Converted waves are neglected. A procedure is suggested for the construction of a depth map of reflectors from observations at the surface. Assuming a velocity model, we may integrate the downgoing wave away from a surface source. Likewise, the upcoming wave may be approximately integrated back down into the earth. Since reflection coefficients are real, the ratio of upcoming to downgoing waves tends to be real at a reflector. An example is given in two dimensions which shows that this ratio over a dipping bed gives the dip correctly independent of source/receiver‐group offset.
Earlier work developed a method of migration of seismic data based on numerical solutions of partial differential equations. The method was designed for the geometry of a single source with a line of surface receivers. Here the method is extended to the geometry of stacked sections, or what is nearly the same thing, to the geometry where a source and receiver move together along the surface as in marine profiling. The basic idea simply stated is that the best receiver line for any reflector is just at (or above) the reflector. Data received at a surface line of receivers may be extrapolated by computer to data at a hypothetical receiver line at any depth. By considering migration before stacking over offset, it is found that certain ambiguities in velocity analysis may be avoided.
A bstract
Claerbout's method has been implemented for the migration of stacked seismic data. A simplified description of the method is given together with an account of some of the practical programming problems and the types of inaccuracy encountered. Routine production results are considered to be comparable or superior to the results derived from alternative migration techniques. Particular advantages are 1) the possibility of using a detailed velocity model for the migration and 2) the preservation of the amplitude and character of the seismic events on the migrated time section.
A bstract
A process is described whereby the interpretation of seismic reflection data is carried out by a preliminary two‐dimensional plotting procedure followed by a three‐dimensional migration. The concept of a surface of maximum convexity is introduced as an integral part of the process of migration. The procedures for deriving the necessary charts of curves are considered and a number of serviceable charts presented.
The complete motion of an elastic quarter plane and of a three-quarter plane with free boundaries caused by an explosive point source, is obtained by finite difference methods.
Varying ratio β/α of the shear to compressional wave velocity shows that in the quarter plane the amplitude of motion at the corner increases with increasing β/α, in the three-quarter plane it decreases. The motion in the quarter plane differs from the sum of reflections at perpendicular half planes. The amplitude of diffracted P waves varies mainly with distance from the corner. The amplitude of diffracted S waves varies mainly in angular direction. Corner-generated surface waves and elliptical particle motion in the waves are analysed. At the corner of a quarter plane, the amplitude of the Rayleigh wave is three to five times as large as on a half plane, the particle motion is elliptic and the major axes of the ellipses are inclined at 45° to the free surface.
A finite-difference technique is described for producing synthetic seismograms for complex subsurface geometries and for arbitrary source-receiver separations. Synthetic seismograms computed for several models of exploration interest serve to illustrate how the technique may help the interpreter. The examples also illustrate various implementational aspects of the finite-difference approach, which involve such phenomena as grid dispersion, artificial reflections from the edge of the model, and choice of spatial and temporal sampling increments. The two-dimensional partial differential equations of motion describing the propagation of stress waves in an elastic medium are approximated by suitable finite-difference equations, which can be solved on a discrete spatial grid by strictly numerical procedures.
Wave equation migration is known to be simpler in principle when the horizontal coordinate or coordinates are replaced by their Fourier conjugates. Two practical migration schemes utilizing this concept are developed in this paper. One scheme extends the Claerbout finite difference method, greatly reducing dispersion problems usually associated with this method at higher dips and frequencies. The second scheme effects a Fourier transform in both space and time; by using the full scalar wave equation in the conjugate space, the method eliminates (up to the aliasing frequency) dispersion altogether. The second method in particular appears adaptable to three-dimensional migration and migration before stack.
Accurate methods for the solution of the migration of zero-offset seismic records have been developed. The numerical operations are defined in the frequency domain. The source and recorder positions are lowered by means of a phase shift, or a rotation of the phase angle of the Fourier coefficients. For applications with laterally invariant velocities, the equations governing the migration process are solved very accurately by the phase-shift method. The partial differential equations considered include the 15 degree equation, as well as higher order approximations to the exact migration process. The most accurate migration is accomplished by using the asymptotic equation, whose dispersion relation is the same as that of the full wave equation for downward propagating waves. These equations, however, do not account for the reflection and transmission effects, multiples, or evanescent waves. For comparable accuracy, the present approach to migration is expected to be computationally more efficient than f
Computer generated seis-mograms Boore, D. M., 1970, Finite-difference solution to the equations of elastic wave propagation with application to Love waves over dip-ping interfaces Coarse grid calculation of waves in inhomoge-neous media with applications to delineation of complicated seismic Roy
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Wave equation migration: two approaches: Presented at the 46th Annual International SEC Meet-ing November 6
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