Elastic Reverse Time Migration Research Articles

Converted-wave guided imaging condition for elastic reverse time migration with wavefield separation

Elastic reverse time migration (ERTM), which is capable of using multicomponent seismic data, provides not only an improvement of the P-P image compared to the one derived from acoustic RTM, but also more opportunities to understand the subsurface through converted wave images (P-S, S-P, and S-S images). However, the polarity reversals in P-S and S-P images and cross-talk noises generated in S-P and S-S images degrade the converted wave images of ERTM. To overcome these problems, we derive a new P-S converted wave imaging condition for 2D ERTM based on wavefield separation techniques. The proposed imaging condition, called converted-wave guided (CWG) imaging condition, incorporates an extra term that represents the sign and wavelength of S-waves converted from source wavefields into the zero-lag cross-correlation imaging condition for P-S imaging. The extra term compensates for the polarity reversal of separated S-waves from receiver wavefields because the converted S-waves from source wavefields also have the change in polarity. In addition, since this CWG imaging condition produces images where P- and S-waves from source wavefields and S-waves from receiver wavefields coincide, image resolution is enhanced without generating spurious events. Our approach is motivated by the specific feature of ERTM that generates converted waves at the reflection points (conventional imaging points) when proper elastic models are used. Through a numerical experiment with a simple elastic model, we demonstrate that the proposed CWG imaging condition successfully corrects the polarity reversal and provides higher image resolution. We also test our migration algorithm on a synthetic ocean bottom cable (OBC) dataset created using the Marmousi-II model. The P-S image obtained from CWG imaging condition shows continuous events and improved image resolution.

ELASTIC REVERSE TIME MIGRATION BASED ON WAVEFIELD SEPARATION

AbstractCompared 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.

Elastic least-squares reverse time migration via linearized elastic full-waveform inversion with pseudo-Hessian preconditioning

Time-domain elastic least-squares reverse time migration (LSRTM) is formulated as a linearized elastic full-waveform inversion problem. The elastic Born approximation and elastic reverse time migration (RTM) operators are derived from the time-domain continuous adjoint-state method. Our approach defines P- and S-wave impedance perturbations as unknown elastic images. Our algorithm is obtained using continuous functional analysis in which the problem is discretized at the final stage (optimize-before-discretize approach). The discretized numerical versions of the elastic Born operator and its adjoint (elastic RTM operator) pass the dot-product test. The conjugate gradient least-squares method is used to solve the least-squares migration quadratic optimization problem. In other words, the Hessian operator for elastic LSRTM is implicitly inverted via a matrix-free algorithm that only requires the action of forward and adjoint operators on vectors. The diagonal of the pseudo-Hessian operator is used to design a preconditioning operator to accelerate the convergence of the elastic LSRTM. The elastic LSRTM provides higher resolution images with fewer artifacts and a superior balance of amplitudes when compared with elastic RTM. More important, elastic LSRTM can mitigate crosstalk between the P- and S-wave impedance perturbations given that the off-diagonal elements of the Hessian are attenuated via the inversion.

3D angle decomposition for elastic reverse time migration

We have developed three approaches for 3D angle decomposition using elastic reverse time migration. The first approach uses time- and space-lag common-image point gathers computed from elastic wavefields. This method facilitates computing angle gathers at sparse and possibly irregularly distributed points in the image. The second approach transforms extended time-lag images to the angle domain using slant stacks along 4D surfaces, instead of using slant stacks along 2D straight lines. The third approach transforms space-lag common-image gathers to the angle domain. The three proposed methods solve a system of equations that handles dipping reflectors, and they yield angle gathers that are more accurate compared with those obtained via alternative existing methods. We have developed our methods using 2D and 3D synthetic and field data examples and found that they provide accurate opening and azimuth angles and they can handle steeply dipping reflectors and converted wave modes.

Elastic least-squares reverse time migration with hybrid l1/l2 misfit function

We have developed the theory and synthetic tests of elastic least-squares reverse time migration (ELSRTM). In this method, a least-squares reverse time migration algorithm is used to image multicomponent seismic data based on the first-order elastic velocity-stress wave equation, in which the linearized elastic modeling equations are used for forward modeling and its adjoint equations are derived based on the adjoint-state method for back propagating the data residuals. Also, we have developed another ELSRTM scheme based on the wavefield separation technique, in which the P-wave image is obtained using P-wave forward and adjoint wavefields and the S-wave image is obtained using P-wave forward and S-wave adjoint wavefields. In this way, the crosstalk artifacts can be minimized to a significant extent. In general, seismic data inevitably contain noise. We apply the hybrid [Formula: see text] misfit function to the ELSRTM algorithm to improve the robustness of our ELSRTM to noise. Numerical tests on synthetic data reveal that our ELSRTM, when compared with elastic reverse time migration, can produce images with higher spatial resolution, more-balanced amplitudes, and fewer artifacts. Moreover, the hybrid [Formula: see text] misfit function makes the ELSRTM more robust than the [Formula: see text] misfit function in the presence of noise.

Anisotropic elastic wavefield imaging using the energy norm

Based on the energy conservation principle, we derive a scalar imaging condition for anisotropic elastic wavefield migration. Compared with conventional imaging conditions that correlate displacement components or potentials from source and receiver wavefields, the proposed imaging condition does not suffer from polarity reversal, which degrades the image quality after stacking over shots. Our imaging condition also accounts for the directionality of the wavefields in space and time, leading to the attenuation of backscattering artifacts, which commonly appear in elastic reverse time migration images in the presence of strong model contrasts. In addition, our new imaging condition does not require wave-mode decomposition, which demands significant additional cost for elastic wavefields in anisotropic media. To properly image structures, we rely on the anisotropy parameters used in migration, as one would do for any other imaging condition. Our imaging condition is suitable for arbitrary anisotropy. We show the successful application of the anisotropic energy imaging condition by performing numerical experiments on simple and complex geologic models. We compare its quality with conventional counterparts by simulating complex geologic settings with vertical or tilted transverse isotropy.

Reverse Time Migration of Elastic Waves Using the Pseudospectral Time-Domain Method

Reverse time migration (RTM), especially that for elastic waves, consumes massive computation resources which limit its wide applications in industry. We suggest to use the pseudospectral time-domain (PSTD) method in elastic wave RTM. RTM using PSTD can significantly reduce the computational requirements compared with RTM using the traditional finite difference time domain method (FDTD). In addition to the advantage of low sampling rate with high accuracy, the PSTD method also eliminates the periodicity (or wraparround) limitation caused by fast Fourier transform in the conventional pseudospectral method. To achieve accurate results, the PSTD method needs only about half the spatial sampling rate of the twelfth-order FDTD method. Thus, the PSTD method can save up to 87.5% storage memory and 90% computation time over the twelfth-order FDTD method. We implement RTM using PSTD for elastic wave equations and accelerate it by Open Multi-Processing technology. To keep the computational load balance in parallel ...

Vector-based elastic reverse time migration based on scalar imaging condition

The scalar images (PP, PS, SP, and SS) of elastic reverse time migration (ERTM) can be generated by applying an imaging condition as crosscorrelation of pure wave modes. In conventional ERTM, Helmholtz decomposition is commonly applied in wavefield separation, which leads to a polarity reversal problem in converted-wave images because of the opposite polarity distributions of the S-wavefields. Polarity reversal of the converted-wave image will cause destructive interference when stacking over multiple shots. Besides, in the 3D case, the curl calculation generates a vector S-wave, which makes it impossible to produce scalar PS, SP, and SS images with the crosscorrelation imaging condition. We evaluate a vector-based ERTM (VB-ERTM) method to address these problems. In VB-ERTM, an amplitude-preserved wavefield separation method based on decoupled elastic wave equation is exploited to obtain the pure wave modes. The output separated wavefields are both vectorial. To obtain the scalar images, the scalar imaging condition in which the scalar product of two vector wavefields with source-normalized illumination is exploited to produce scalar images instead of correlating Cartesian components or magnitude of the vector P- and S-wave modes. Compared with alternative methods for correcting the polarity reversal of PS and SP images, our ERTM solution is more stable and simple. Besides these four scalar images, the VB-ERTM method generates another PP-mode image by using the auxiliary stress wavefields. Several 2D and 3D numerical examples are evaluated to demonstrate the potential of our ERTM method.

Elastic least-squares reverse time migration

We use elastic least-squares reverse time migration (LSRTM) to invert for the reflectivity images of P- and S-wave impedances. Elastic LSRTM solves the linearized elastic-wave equations for forward modeling and the adjoint equations for backpropagating the residual wavefield at each iteration. Numerical tests on synthetic data and field data reveal the advantages of elastic LSRTM over elastic reverse time migration (RTM) and acoustic LSRTM. For our examples, the elastic LSRTM images have better resolution and amplitude balancing, fewer artifacts, and less crosstalk compared with the elastic RTM images. The images are also better focused and have better reflector continuity for steeply dipping events compared to the acoustic LSRTM images. Similar to conventional least-squares migration, elastic LSRTM also requires an accurate estimation of the P- and S-wave migration velocity models. However, the problem remains that, when there are moderate errors in the velocity model and strong multiples, LSRTM will produce migration noise stronger than that seen in the RTM images.

Elastic wavefield separation based on the Helmholtz decomposition

Divergence and curl operators used for the decomposition of P- and S-wave modes in elastic reverse time migration (RTM) change the amplitudes, units, and phases of extrapolated wavefields. I separate the P- and S-waves in elastic media based on the Helmholtz decomposition. The decomposed wavefields based on this approach have the same amplitudes, units, and phases as the extrapolated wavefields. To avoid expensive multidimensional integrals in the Helmholtz decomposition, I introduce a fast Poisson solver to efficiently solve the vector Poisson’s equation. This fast algorithm allows us to reduce computational complexity from [Formula: see text] to [Formula: see text], where [Formula: see text] is the total number of grid points. Because the decomposed P- and S-waves are vector fields, I use vector imaging conditions to construct PP-, PS-, SS-, and SP-images. Several 2D numerical examples demonstrate that this approach allows us to accurately and efficiently decompose P- and S-waves in elastic media. In addition, elastic RTM images based on the vector imaging conditions have better quality and avoid polarity reversal in comparison with images based on the divergence and curl separation or direct component-by-component crosscorrelation.

Viscoelastic reverse time migration with attenuation compensation

We have developed a theory of viscoelastic reverse time migration (RTM). The main feature of viscoelastic RTM is a compensation for P- and S-wave attenuation effects in seismic images during migration. The forward modeling engine is based on a viscoelastic wave equation involving fractional Laplacians. Because of the decoupled attenuation property, wave propagation can be simulated in three scenarios, i.e., only the amplitude loss effect, only the phase dispersion effect, or both effects simultaneously. This separation brings practical flexibility to studying attenuation effects on wave propagation and imaging. The backward modeling operator is constructed by reversing the sign of first-order time derivative amplitude loss operators. Synthetic examples determine the ability of viscoelastic RTM to illuminate degraded areas and shadow zones caused by attenuation. Numerical experiments also reveal that [Formula: see text]-compensated imaging is noticeably more accurate in kinematics and dynamics than elastic imaging in the presence of high attenuation. Results from a synthetic 3D model determine the superiority of viscoelastic RTM over elastic RTM in imaging salt flanks and delineation of salt boundaries, which are dimmed in elastic images.

Converted-wave seismic imaging: Amplitude-balancing source-independent imaging conditions

We have developed crosscorrelational and deconvolutional forms of a source-independent converted-wave imaging condition (SICW-IC) and show the relationship between them using a concept of conversion ratio coefficient, a concept that we developed through reflection, transmission, and conversion coefficients. We applied the SICW-ICs to a two half-space model and the synthetic Marmousi I and II models and show the sensitivity of the SICW-ICs to incorrect wave speed models. We also compare the SICW-ICs and source-dependent elastic reverse time migration. The results of SICW-ICs highlight the improvements in spatial resolution and amplitude balancing with the deconvolutional forms. This is an attractive alternative to active and passive source elastic imaging.

Separating quasi-P-wave in transversely isotropic media with a vertical symmetry axis by synthesized pressure applied to ocean-bottom seismic data elastic reverse time migration

Quasi-P (qP)-wavefield separation is a crucial step for elastic P-wave imaging in anisotropic media. It is, however, notoriously challenging to quickly and accurately obtain separated qP-wavefields. Based on the concepts of the trace of the stress tensor and the pressure fields defined in isotropic media, we have developed a new method to rapidly separate the qP-wave in a transversely isotropic medium with a vertical symmetry axis (VTI) by synthesized pressure from ocean-bottom seismic (OBS) data as a preprocessing step for elastic reverse time migration (ERTM). Another key aspect of OBS data elastic wave imaging is receiver-side 4C records back extrapolation. Recent studies have revealed that receiver-side tensorial extrapolation in isotropic media with ocean-bottom 4C records can sufficiently suppress nonphysical waves produced during receiver-side reverse time wavefield extrapolation. Similarly, the receiver-side 4C records tensorial extrapolation was extended to ERTM in VTI media in our studies. Combining a separated qP-wave by synthesizing pressure and receiver-side wavefield reverse time tensorial extrapolation with the crosscorrelation imaging condition, we have developed a robust, fast, flexible, and elastic imaging quality improved method in VTI media for OBS data.

Elastic-wave reverse-time migration based on decoupled elastic-wave equations and inner-product imaging condition

Polarity reversal in converted wave images of elastic reverse time migration destructs the reflection events after stacking multi-shot migration profile. We derive a new imaging method for elastic reverse-time migration to automatically circumvent polarity reversal. Instead of obtaining scalar P-wave and vector S-wave potentials from the wavefield by using Helmholtz decomposition as in conventional methods, we obtain vector P- and S-wave displacement wavefields based on a decoupled elastic wave equation, which denotes the displacement component along the propagation direction and perpendicular to the propagation direction, respectively. Based on this decomposition method, the vector P- and S-wavefields preserve the amplitude and phase attributes of the original wavefield. As for the vector wavefields (vector P- and S-wave displacement wavefields), the inner-product imaging condition is proposed to extract reflectivity of specified wave modes at interfaces. The analysis of the imaging kernel demonstrates this imaging condition is valid not only for pure-mode imaging (PP and SS), but also for converted wave imaging (PS and SP) of ground-based seismic exploration. With this new method, we do not have to correct the polarity reversal in converted wave images, which is an essential step in the conventional method with expensive computation costs. Numerical examples with synthetic data have shown that the inner product imaging method works and the quality of the imaging events is effectively improved.

Implementation of Elastic Prestack Reverse-Time Migration Using an Efficient Finite-Difference Scheme

Elastic reverse-time migration (RTM) can reflect the underground elastic information more comprehensively than single-component Pwave migration. One of the most important requirements of elastic RTM is to solve wave equations. The imaging accuracy and efficiency of RTM depends heavily on the algorithms used for solving wave equations. In this paper, we propose an efficient staggered-grid finite-difference (SFD) scheme based on a sampling approximation method with adaptive variable difference operator lengths to implement elastic prestack RTM. Numerical dispersion analysis and wavefield extrapolation results show that the sampling approximation SFD scheme has greater accuracy than the conventional Taylor-series expansion SFD scheme. We also test the elastic RTM algorithm on theoretical models and a field data set, respectively. Experiments presented demonstrate that elastic RTM using the proposed SFD scheme can generate better images than that using the Taylor-series expansion SFD scheme, particularly for PS images. FurH. thermore, the application of adaptive variable difference operator lengths can effectively improve the computational efficiency of elastic RTM.

Acoustic-elastic coupled equation for ocean bottom seismic data elastic reverse time migration

Conventionally, multicomponent geophones used to record the elastic wavefields in the solid seabed are necessary for ocean bottom seismic (OBS) data elastic reverse time migration (RTM). Particle velocity components are usually injected directly as boundary conditions in the elastic-wave equation in the receiver-side wavefield extrapolation step, which causes artifacts in the resulting elastic images. We have deduced a first-order acoustic-elastic coupled equation (AECE) by substituting pressure fields into the elastic velocity-stress equation (EVSE). AECE has three advantages for OBS data over EVSE when performing elastic RTM. First, the new equation unifies wave propagation in acoustic and elastic media. Second, the new equation separates P-waves directly during wavefield propagation. Third, three approaches are identified when using the receiver-side multicomponent particle velocity records and pressure records in elastic RTM processing: (1) particle velocity components are set as boundary conditions in receiver-side vectorial extrapolation with the AECE, which is equal to the elastic RTM using the conventional EVSE; (2) the pressure component may also be used for receiver-side scalar extrapolation with the AECE, and with which we can accomplish PP and PS images using only the pressure records and suppress most of the artifacts in the PP image with vectorial extrapolation; and (3) ocean-bottom 4C data can be simultaneously used for elastic images with receiver-side tensorial extrapolation using the AECE. Thus, the AECE may be used for conventional elastic RTM, but it also offers the flexibility to obtain PP and PS images using only pressure records.

Scalar and vector imaging based on wave mode decoupling for elastic reverse time migration in isotropic and transversely isotropic media

Recording P- and S-wave modes acquires more information related to rock properties of the earth’s interior. Elastic migration, as a part of multicomponent seismic data processing, potentially offers a great improvement over conventional acoustic migration to create a spatial image of some medium properties. In the framework of elastic reverse time migration, we have developed new scalar and vector imaging conditions assisted by efficient polarization-based mode decoupling to avoid crosstalk among the different wave modes for isotropic and transversely isotropic media. For the scalar imaging, we corrected polarity reversal of zero-lag PS images using the local angular attributes on the fly of angle-domain imaging. For the vector imaging, we naturally used the polarization information in the decoupled single-mode vector fields to automatically avoid the polarity reversal and to estimate the local angular attributes for angle-domain imaging. Examples of increasing complexity in 2D and 3D cases found that the proposed approaches can be used to obtain a physically interpretable image and angle-domain common-image gather at an acceptable computational cost. Decoupling and imaging the 3D S-waves involves some complexity, which has not been addressed in the literature. For this reason, we also attempted at illustrating the physical contents of the two separated S-wave modes and their contribution to seismic full-wave imaging.

Elastic reverse-time migration based on amplitude-preserving P- and S-wave separation

Imaging the PP- and PS-wave for the elastic vector wave reverse-time migration requires separating the P- and S-waves during the wave field extrapolation. The amplitude and phase of the P- and S-waves are distorted when divergence and curl operators are used to separate the P- and S-waves. We present a P- and S-wave amplitude-preserving separation algorithm for the elastic wavefield extrapolation. First, we add the P-wave pressure and P-wave vibration velocity equation to the conventional elastic wave equation to decompose the P- and S-wave vectors. Then, we synthesize the scalar P- and S-wave from the vector Pand S-wave to obtain the scalar P- and S-wave. The amplitude-preserved separated P- and S-waves are imaged based on the vector wave reverse-time migration (RTM). This method ensures that the amplitude and phase of the separated P- and S-wave remain unchanged compared with the divergence and curl operators. In addition, after decomposition, the P-wave pressure and vibration velocity can be used to suppress the interlayer reflection noise and to correct the S-wave polarity. This improves the image quality of P- and S-wave in multicomponent seismic data and the true-amplitude elastic reverse time migration used in prestack inversion.

Up/down and P/S decompositions of elastic wavefields using complex seismic traces with applications to calculating Poynting vectors and angle-domain common-image gathers from reverse time migrations

Separations of up- and down-going as well as of P- and S-waves are often a part of processing of multicomponent recorded data and propagating wavefields. Most previous methods for separating up/down propagating wavefields are expensive because of the requirement to save time steps to perform Fourier transforms over time. An alternate approach for separation of up-and down-going waves, based on extrapolation of complex data traces is extended from acoustic to elastic, and combined with P- and S-wave decomposition by decoupled propagation. This preserves all the information in the original data and eliminates the need for a Fourier transform over time, thereby significantly reducing the storage cost and improving computational efficiency. Wavefield decomposition is applied to synthetic elastic VSP data and propagating wavefield snapshots. Poynting vectors obtained from the particle velocity and stress fields after P/S and up/down decompositions are much more accurate than those without because interference between the corresponding wavefronts is significantly reduced. Elastic reverse time migration with the P/S and up/down decompositions indicated significant improvement compared with those without decompositions, when applied to elastic data from a portion of the Marmousi2 model.

Isotropic elastic wavefield imaging using the energy norm

From the elastic-wave equation and the energy conservation principle, we have derived an energy norm that is applicable to imaging with elastic wavefields. Extending the concept of the norm to an inner product enables us to compare two related wavefields. For example, the inner product of source and receiver wavefields at each spatial location leads to an imaging condition. This new imaging condition outputs a single image representing the total reflection energy, and it contains individual terms related to the kinetic and potential energy (strain energy) from both extrapolated wavefields. An advantage of the proposed imaging condition compared with alternatives is that it does not suffer from polarity reversal at normal incidence, as do conventional images obtained using converted waves. Our imaging condition also accounted for the directionality of the wavefields in space and time. Based on this information, we have modified the imaging condition for attenuation of backscattering artifacts in elastic reverse time migration images. We performed numerical experiments that revealed the improved quality of the energy images compared with their conventional counterparts and the effectiveness of the imaging condition in attenuating backscattering artifacts even in media characterized by high spatial variability.