Source Wavefields Research Articles

Common-image gathers using the excitation amplitude imaging condition

Common-image gathers (CIGs) are extensively used in migration velocity analysis. Any defocused events in the subsurface offset domain or equivalently nonflat events in angle-domain CIGs are accounted for revising the migration velocities. However, CIGs from wave-equation methods such as reverse time migration are often expensive to compute, especially in 3D. Using the excitation amplitude imaging condition that simplifies the forward-propagated source wavefield, we have managed to extract extended images for space and time lags in conjunction with prestack reverse time migration. The extended images tend to be cleaner, and the memory cost/disk storage is extensively reduced because we do not need to store the source wavefield. In addition, by avoiding the crosscorrelation calculation, we reduce the computational cost. These features are demonstrated on a linear [Formula: see text] model, a two-layer velocity model, and the Marmousi model.

Analysis of up/down decomposed acoustic reverse time migration images

With the up/down separation of propagating wavefields using complex traces, (forward time extrapolated) source and (reverse time extrapolated) receiver wavefields are decomposed into up- and downgoing parts in acoustic reverse time migration (RTM). Together with the crosscorrelation imaging condition, four images (down-up, up-down, up-up, and down-down) are generated, which facilitate the analysis of artifacts and the imaging ability of the four images. Artifacts may exist in all of the decomposed images, but their positions and types are different. We have explained and illustrated the causes of artifacts in different images with sketches and numerical tests. The up-down image better images subsalt turned waves than does the down-up image; thus, decomposed images have advantages over a single composite image for target interpretation.

Polarized wavefield magnitudes with optical flow for elastic angle-domain common-image gathers

We have developed a polarized wavefield magnitude (PWM) approach to determine the polarity of an elastic vector wavefield magnitude, for elastic prestack reverse time migration imaging and common-image gather generation. Explicit decomposition of the coupled elastic wavefield into separate P-waves and converted S-waves is performed by using decoupled elastodynamic wavefield propagation. Stable source and receiver wavefield propagation angles are determined by an optical flow method for elastic media. The sign ambiguity in the propagation directions for incident P- and reflected P- and S-wavefield vector magnitudes is resolved by establishing a relation between the propagation and particle velocity vectors. From the computed propagation angles, PP- and PS-wavefields are imaged using source-normalized crosscorrelation followed by sorting into common-image gathers. Modeling of amplitude variations with angle (AVA) from single-shot migrations and from elastic angle-domain common-image gathers demonstrates that the amplitude fidelity is maintained. Mode-converted PS reflectivities also give an independent set of accurate AVA information. Numerical results for the elastic Marmousi2 model confirm PWM as a physically valid and robust method for elastic wavefield imaging in arbitrarily complicated media.

Acoustic wavefield imaging using the energy norm

Wavefield energy can be measured by the so-called energy norm. We have extended the concept of “norm” to obtain the energy inner product between two related wavefields. Considering an imaging condition as an inner product between the source and receiver wavefields at each spatial location, we have developed a new imaging condition that represents the total reflection energy. Investigating this imaging condition further, we have found that it accounts for wavefield directionality in space time. Based on the directionality discrimination provided by this imaging condition, we have applied it to attenuate backscattering artifacts in reverse time migration (RTM). This imaging condition can be designed not only to attenuate backscattering artifacts, but also to attenuate any selected reflection angle. By exploiting the flexibility of this imaging condition for attenuating certain angles, we have developed a procedure to preserve the type of events that propagate along the same path, i.e., backscattered, diving, and head waves, leading to a suitable application for full-waveform- inversion (FWI). This application involves filtering the FWI gradient to preserve the tomographic term (waves propagating in the same path) and attenuate the migration term (reflections) of the gradient. We have developed the energy imaging condition applications for RTM and FWI using numerical experiments in simple (horizontal reflector) and complex models (Sigsbee and Marmousi).

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.

Accelerating extended least-squares migration with weighted conjugate gradient iteration

Least-squares migration (LSM) iteratively achieves a mean-square best fit to seismic reflection data, provided that a kinematically accurate velocity model is available. The subsurface offset extension adds extra degrees of freedom to the model, thereby allowing LSM to fit the data even in the event of significant velocity error. This type of extension also implies additional computational expense per iteration from crosscorrelating source and receiver wavefields over the subsurface offset, and therefore places a premium on rapid convergence. We have accelerated the convergence of extended least-squares migration by combining the conjugate gradient algorithm with weighted norms in range (data) and domain (model) spaces that render the extended Born modeling operator approximately unitary. We have developed numerical examples that demonstrate that the proposed algorithm dramatically reduces the number of iterations required to achieve a given level of fit or gradient reduction compared with conjugate gradient iteration with Euclidean (unweighted) norms.

Subsurface reflectivity estimation from imaging of primaries and multiples using amplitude-normalized separated wavefields

Several studies have shown the benefits of including multiple reflections together with primaries in the structural imaging of subsurface reflectors. However, to characterize the reflector properties, there is a need to compensate for propagation effects due to multiple scattering and to properly combine the information from primaries and all orders of multiples. From this perspective and based on the wave equation and Rayleigh’s reciprocity theorem, recent works have suggested computing the subsurface image from the Green’s function reflection response (or reflectivity) by inverting a Fredholm integral equation in the frequency-space domain. By following Claerbout’s imaging principle and assuming locally reacting media, the integral equation may be reduced to a trace-by-trace deconvolution imaging condition. For a complex overburden and considering that the structure of the subsurface is angle-dependent, this trace-by-trace deconvolution does not properly solve the Fredholm integral equation. We have inverted for the subsurface reflectivity by solving the matrix version of the Fredholm integral equation at every subsurface level, based on a multidimensional deconvolution of the receiver wavefields with the source wavefields. The total upgoing pressure and the total filtered downgoing vertical velocity were used as receiver and source wavefields, respectively. By selecting appropriate subsets of the inverted reflectivity matrix and by performing an inverse Fourier transform over the frequencies, the process allowed us to obtain wavefields corresponding to virtual sources and receivers located in the subsurface, at a given level. The method has been applied on two synthetic examples showing that the computed reflectivity wavefields are free of propagation effects from the overburden and thus are suited to extract information of the image point location in the angular and spatial domains. To get the computational cost down, our approach is target-oriented; i.e., the reflectivity may only be computed in the area of most interest.

Eliminating the redundant source effects from the cross-correlation reverse-time migration using a modified stabilized division

Cross-correlation reverse-time migration is the kernel of two-way wave-equation migration and inversion. However, it more or less tapers the spectrum of receiver data due to a redundant overlay of the source wavelet, whose amplitude spectrum is usually bandlimited and non-flat. To circumvent this issue, there are two optional strategies: whitening the source directly, or preconditioning the seismic traces by division with the amplitude spectrum of the source in the frequency domain. In this paper, we choose the latter one because the source signature is crucial to illumination compensation and seismic inversion. To avoid division by zero, a modified stabilized division algorithm based on the Taylor-expansion is developed. The modified division is easy to complete with computers and can be extend to any order. Moreover, when simulating 2-D source wavefield, the half-integral effect is also considered. We will demonstrate our proposed scheme using the Sigsbee2b synthetic data and a real field data.

Least-squares reverse-time migration with cost-effective computation and memory storage

Least-squares reverse-time migration (LSRTM), which involves several iterations of reverse-time migration (RTM) and Born modeling procedures, can provide subsurface images with better balanced amplitudes, higher resolution and fewer artifacts than standard migration. However, the same source wavefield is repetitively computed during the Born modeling and RTM procedures of different iterations. We developed a new LSRTM method with modified excitation-amplitude imaging conditions, where the source wavefield for RTM is forward propagated only once while the maximum amplitude and its excitation-time at each grid are stored. Then, the RTM procedure of different iterations only involves: (1) backward propagation of the residual between Born modeled and acquired data, and (2) implementation of the modified excitation-amplitude imaging condition by multiplying the maximum amplitude by the back propagated data residuals only at the grids that satisfy the imaging time at each time-step. For a complex model, 2 or 3 local peak-amplitudes and corresponding traveltimes should be confirmed and stored for all the grids so that multiarrival information of the source wavefield can be utilized for imaging. Numerical experiments on a three-layer and the Marmousi2 model demonstrate that the proposed LSRTM method saves huge computation and memory cost.

Drill-rig noise suppression using the Karhunen-Loéve transform for seismic-while-drilling experiment at Brukunga, South Australia

Diamond-impregnated drill bits are known to be low energy vibration seismic sources. With the strong interference from the drill rig, it is difficult to obtain the drill-bit wavefield with a surface receiver array. To overcome the challenge of surface wave interference generated from the rig for seismic-while-drilling (SWD), we need to separate the rig- and bit-generated signals. To this end, we apply two wavefield separation methods, the Karhunen-Loéve (KL) transform and the f – k filter, and compare their performance. The applicability of these methods is based on the drill rig and drill bit having different spatial positions. While the drill-bit spatial position changes during the process of drilling, the drill rig remains stationary. This results in the source wavefields from the drill rig and the drill-bit having different characteristics, and allows us to separate and extract the drill-bit signal. We use a synthetic model to compare the KL transform and f – k filter. Both techniques are robust when the noise wavefield has consistent amplitude moveout. However, for changing amplitudes, such as the rig noise, which has an unrepeatable wavefield due to power amplitude variation, we show that the KL transform performs better in such situations. We also show the results of signal analysis of the SWD experiment data acquired from Brukunga, South Australia. We demonstrate the feasibility of the KL transform in separating the coherent noises from the stationary drill rig in a hard rock drilling environment, particularly emphasising the suppression of the surface and direct waves from the rig. The results show that drill-rig noise can be effectively suppressed in the correlation domain.

Reverse time migration of 3D vertical seismic profile data

Reverse time migration (RTM) has shown increasing advantages in handling seismic images of complex subsurface media, but it has not been used widely in 3D seismic data due to the large storage and computation requirements. Our prime objective was to develop an RTM strategy that was applicable to 3D vertical seismic profiling (VSP) data. The strategy consists of two aspects: storage saving and calculation acceleration. First, we determined the use of the random boundary condition (RBC) to save the storage in wavefield simulation. An absorbing boundary such as the perfect matching layer boundary is often used in RTM, but it has a high memory demand for storing the source wavefield. RBC is a nonabsorbing boundary and only stores the source wavefield at the two maximum time steps, then repropagates the source wavefield backwards at every time step, and hence, it significantly reduces the memory requirement. Second, we examined the use of the graphic processing unit (GPU) parallelization technique to accelerate the computation. RBC needs to simulate the source wavefield twice and doubles the computation. Thus, it is very necessary to realize the RTM algorithm by GPU, especially for a 3D VSP data set. GPU and central processing unit (CPU) collaborated parallel implementation can greatly reduce the computation time, where the CPU performs serial code, and the GPU performs parallel code. Because RBC does not need the same huge amount of storage as an absorbing boundary, RTM becomes practically applicable for 3D VSP imaging.

Angle gathers from reverse time migration using analytic wavefield propagation and decomposition in the time domain

Angle-domain common imaging gathers (ADCIGs) are important input data for migration velocity analysis and amplitude variation with angle analysis. Compared with Kirchhoff migration and one-way wave equation migration, reverse time migration (RTM) is the most accurate imaging method in complex areas, such as the subsalt area. We have developed a method to generate ADCIGs from RTM using analytic wavefield propagation and decomposition. To estimate the wave-propagation direction and angle by spatial Fourier transform during the time domain wave extrapolation, we have developed an analytic wavefield extrapolation method. Then, we decomposed the extrapolated source and receiver wavefields into their local angle components (i.e., local plane-wave components) and applied the angle-domain imaging condition to form ADCIGs. Because the angle-domain imaging condition is a convolution imaging condition about the source and receiver propagation angles, it is costly. To increase the efficiency of the angle-domain imaging condition, we have developed a local plane-wave decomposition method using matching pursuit. Numerical examples of synthetic and real data found that this method could generate high-quality ADCIGs. And these examples also found that the computational cost of this approach was related to the complexity of the source and receiver wavefields.

Reverse-time prestack depth migration of GPR data from topography for amplitude reconstruction in complex environments

With increased computational power, reverse-time prestack depth migration (RT-PSDM) has become a preferred imaging tool in seismic exploration, yet its use has remained relatively limited in ground-penetrating radar (GPR) applications. Complex topography alters the wavefield kinematics making for a challenging imaging problem. Model simulations show that topographic variation can substantially distort reflection amplitudes due to irregular wavefield spreading, attenuation anomalies due to irregular path lengths, and focusing and defocusing effects at the surface. The effects are magnified when the topographic variations are on the same order as the depth of investigation—a situation that is often encountered in GPR investigations. Here, I use a full wave-equation RT-PSDM algorithm to image GPR data in the presence of large topographic variability relative to the depth of investigation. The source and receiver wavefields are propagated directly from the topographic surface and this approach inherently corrects for irregular kinematics, spreading and attenuation. The results show that when GPR data are acquired in areas of extreme topography, RT-PSDM can accurately reconstruct reflector geometry as well as reflection amplitude.

Reducing data storage in reverse time migration

Prestack reverse time migration (RTM) requires extensive data storage since it computes wavefields in forward time and accesses wavefields in reverse order. We first review several successful schemes that have been proposed to reduce data storage, but require more computational redundancies. We propose two effective strategies to reduce data storage during RTM. The first strategy is based on the Nyquist sampling theorem, which involves no extra computational cost. The fact is that the time sampling intervals required by numerical algorithms or given by field records is generally several times smaller than that satisfied by the Nyquist sampling theorem. Therefore, we can correlate the source wavefields with the receiver wavefields at the Nyquist time step, which helps decrease storage of time history. The second strategy is based on a lossless compression algorithm, which is widely used in computer science and information theory. The compression approach reduces storage significantly at a little computational cost. Numerical examples show that the two proposed strategies are effective and efficient.

Source wavefield reconstruction using a linear combination of the boundary wavefield in reverse time migration

We have developed a new scheme with high accuracy and parsimonious memory requirements to reconstruct the source wavefield in reverse time migration (RTM). This scheme used a linear combination of the boundary wavefield in several layers to reconstruct the wavefield in the imaging domain. The value of the linear combination was stored in one buffer, making the computer memory requirement of our method equal to that of a method that only stored one layer of the boundary wavefield. The coefficients for the linear combination were determined by an optimization technique that minimized the finite difference error in the Fourier domain. The optimal coefficients were presented for finite-difference (FD) stencils of 5–15 grid points. The numerical error of our scheme was analyzed and compared with that of standard FD stencils, representing the conventional method that used multiple layers of boundary wavefield to back propagate the source wavefield. The accuracy of our method is only less accurate than the conventional method in theoretical analysis. However, the storage requirement of our method is merely 1/N of the conventional method if a 2N+1 grid-point FD stencil is used in the space. We have also extended the comparison to two other methods, a one-layer method and an extrapolation method, beyond the conventional method. The numerical results demonstrated that our method can accurately generate high-accuracy images in RTM.

Vector-based elastic reverse time migration

Prestack elastic reverse time migration (RTM) of multicomponent seismic data requires separating PP and PS reflections before, or as part of, applying the image condition, and using image conditions that preserve the angle and amplitude information. Both of these requirements are best achieved when all operations are on vectors. We have created a new 2D migration context for isotropic, elastic RTM, which included decomposition of the elastic source and receiver wavefields into P- and S-wave vectors by decoupled elastodynamic extrapolation, which retained the same stress and particle velocity components as the input data. Then, the propagation directions of the incident and reflected P- and S-waves were calculated directly from the stress and particle velocity definitions of the P- and S-wave Poynting vectors. An excitation-amplitude image condition that scaled the receiver wavelet by the source vector magnitude produced angle-dependent images of PP and PS reflection coefficients with the correct polarities, polarization, and amplitudes. It thus simplified the process of obtaining PP and PS angle-domain common-image gathers (ADCIGs); it was less effort to generate ADCIGs from vector data than from scalar data. We found that the resulting prestack elastic images were nearly identical to the corresponding source-normalized crosscorrelation images and had improved resolution because the wavelet broadening that resulted from the crosscorrelation was not present.

Removing false images in reverse time migration: The concept of de-primary

Primary reflections, like multiples, can generate false images when reverse time migration (RTM) algorithms are used. The false images are formed by the zero-lag correlation of the source wavefields and primary reflections, which are propagated by the migration algorithm along nonphysical paths. These paths are generated by strong velocity gradients or reflection interfaces when the two-way wave equation is used. Conceptually, this type of artifact can be removed by separating up- and downgoing waves, but such separation may be impractical because it often requires storing the entire wavefields at all time steps. We have developed a de-primary RTM method in which such separation can be accomplished without saving the wavefields. The computational cost of the proposed method was only approximately 33% higher than that of conventional RTM algorithms. Using field and synthetic data sets, we have demonstrated the existence of this endemic RTM problem and verified the effectiveness of the de-primary RTM technique for removing the false events.

Understanding the reverse time migration backscattering: noise or signal?

ABSTRACTReverse time migration backscattered events are produced by the cross‐correlation between waves reflected from sharp interfaces (e.g., salt bodies). These events, along with head waves and diving waves, produce the so‐called reverse time migration artefacts, which are visible as low wavenumber energy on migrated images. Commonly, these events are seen as a drawback for the reverse time migration method because they obstruct the image of the geologic structure, which is the real objective for the process. In this paper, we perform numeric and theoretical analysis to understand the reverse time migration backscattering energy in conventional and extended images. We show that the reverse time migration backscattering contains a measure of the synchronization and focusing information between the source and receiver wavefields. We show that this synchronization and focusing information is sensitive to velocity errors; this implies that a correct velocity model produces reverse time migration backscattering with maximum energy. Therefore, before filtering the reverse time migration backscattered energy, we should try to obtain a model that maximizes it.

The AWWE-based hybrid absorbing boundary condition for finite-difference modeling and its application in reverse-time migration

The reverse-time migration (RTM) crosscorrelation imaging condition requires that the forward-propagated source wavefield and the backward-propagated receiver wavefield must be obtained at the same time. The easiest way to get the source wavefield is to save the entire time history of the full wavefield into computer memory. However, this strategy requires huge amount of data storage. It is impossible for large-scale 3D RTM. To reduce the computer memory cost, the back-propagated source wavefield is reconstructed by using the stored boundary wavefield. Its computer memory is proportional to the saved boundary grid points. For high order of spatial finite-difference (FD) schemes, more boundary grid points are needed to be stored, which consumes a large amount of the computer memory required for RTM. To further reduce the computer memory cost, we adopt the hybrid absorbing boundary condition (ABC) combined with the arbitrarily wide-angle wave equations (AWWEs). In our method, three boundary grid points can obtain good absorption. The source wavefield can be accurately reconstructed by using these points and the mirror-image symmetry method. Numerical experiments demonstrate the correctness and effectiveness of the proposed method. We compared our method with the conventional hybrid ABC method based on the 15°one way wave equations (OWWEs). Comparisons show that our method with three boundary grid points can achieve the same absorption as the conventional method with ten boundary grid points. For twentieth order of accuracy in space, our method uses only about 30% of memory requirement and about 59% of computation time required by the conventional method.

Reverse time migration of internal multiples for subsalt imaging

Internal multiple reflections have different propagation paths than primary reflections and surface-related multiples; thus, they can complement illumination where other waves are unavailable. Consequently, migration of internal multiples may provide vital images in the shadow zone of primary reflections. We have developed an approach to image structures with internal multiples. With this method, internal multiples can be separated and used as data for reverse time migration. Instead of applying the imaging condition directly to the source and receiver wavefields, we decomposed the extrapolated source and receiver waves into up- and downgoing waves. Then we applied the imaging condition to upgoing source and receiver waves to form an up-up (UU) image and also to downgoing source and receiver waves to form a down-down (DD) image. The image of the true reflector and the artifacts behave differently in the DD and UU images. A similarity analysis is conducted to the UU and DD images to separate true images from artifacts. Numerical examples with simple velocity models are used to demonstrate how to construct UU and DD imaging conditions and eliminate migration artifacts. Finally, we test the 2D SMAART JV model with salt structures for migration using internal multiples.