Wave-equation Migration Velocity Analysis Research Articles

Full-waveform inversion by model extension: Theory, design, and optimization

We describe a new method, full-waveform inversion by model extension (FWIME), that recovers accurate acoustic subsurface velocity models from seismic data when conventional methods fail. We leverage the advantageous convergence properties of wave equation migration velocity analysis (WEMVA) with the accuracy and high-resolution nature of acoustic full-waveform inversion (FWI) by combining them into a robust mathematically consistent workflow with minimal need for user inputs. The novelty of FWIME resides in the design of a new cost function and a novel optimization strategy to combine the two techniques, making our approach more efficient and powerful than applying them sequentially. We observe that FWIME mitigates the need for accurate initial models and low-frequency long-offset data, which can be challenging to acquire. Our new objective function contains two components. First, we modify the forward mapping of the FWI problem by adding a data-correcting term computed with an extended demigration operator, whose goal is to ensure phase matching between predicted and observed data, even when the initial model is inaccurate. The second component, which is a modified WEMVA cost function, allows us to progressively remove the contributions of the data-correcting term throughout the inversion process. The coupling between the two components is handled by the variable projection method, which reduces the number of adjustable hyperparameters, thereby making our solution simple to use. We devise a model-space multiscale optimization scheme by reparameterizing the velocity model on spline grids to control the resolution of the model updates. We generate three cycle-skipped 2D synthetic data sets, each containing only one type of wave (transmitted, reflected, or refracted), and we analyze how FWIME successfully recovers accurate solutions with the same procedure for all three cases. In a second paper, we apply FWIME to challenging realistic examples where we simultaneously invert all wave modes.

Full-waveform inversion by model extension: Practical applications

Producing reliable acoustic subsurface velocity models still remains the main bottleneck of the oil and gas industry’s traditional imaging sequence. In complex geologic settings, the output of conventional ray-based or wave-equation-based tomographic methods may not be sufficiently accurate for full-waveform inversion (FWI) to converge to a geologically satisfactory earth model. We create a new method referred to as full-waveform inversion by model extension (FWIME) in which a wave-equation migration velocity analysis (WEMVA) technique is efficiently paired with a modified version of FWI. We find that our method is more powerful than applying WEMVA and FWI sequentially, and that it can converge to accurate solutions without the use of a good initial guess or low-frequency energy. We determine FWIME’s potential on five realistic and challenging numerical examples that simulate complex geologic scenarios often encountered in hydrocarbon exploration. We guide the reader step by step throughout the optimization process. We find that our method can simultaneously invert all wave types with the same simple mechanism and without the need for a user-intensive hyperparameter tuning process. In an open-source online repository, we provide our C++/compute unified device architecture (CUDA) numerical implementation accelerated with graphics processing units, encapsulated in a Python interface. All the numerical examples developed are accessible through Python notebooks and fully reproducible.

Wave-equation migration velocity analysis via the optimal-transport-based objective function

Wave-equation migration velocity analysis builds a kinematically accurate macrovelocity model containing the large-scale structures of the model for seismic imaging or full-waveform inversion (FWI). Differential semblance optimization formulates the misfit function in the subsurface domain by applying a penalty to the unfocused subsurface-offset gathers or the nonflattened angle gathers. Such a penalty leads to gradients with strong artifacts and spurious oscillations, which then leads to slow convergence. We formulate the misfit function using the optimal transport (OT) among neighboring traces in angle-domain common-image gathers (CIGs). Specifically, we measure the unflatness of the gathers by comparing the Wasserstein distance of adjacent traces. Because CIGs are ideal to form a probability distribution, which is required by OT, it is natural to use OT to measure the time shifts between two distributions. Our objective function exhibits well-behaved convex and unimodal properties in a simple model with a single interface, and it is expected to be minimum for the correct velocity when the angle gathers are flat. A synthetic data set example on the Marmousi model indicates the ability of our method to provide an initial model for FWI that allows FWI to converge. We also apply the approach on a marine field data set to further demonstrate its effectiveness in estimating the macrovelocity model.

Joint inversion of the reflectivity and the velocity model

During linearized waveform inversion (LWI), the presence of small inaccuracies in the background subsurface model can lead to unfocused seismic events in the final image. The effect on the amplitude can mislead the interpretation. We have developed a joint inversion scheme in the model domain of the reflectivity and the background velocity model. The idea is to unify the inversion of the background and the reflectivity model into a single framework instead of treating them as decoupled problems. We find that, with this method, we can obtain a better estimate of the reflectivity than that obtained with conventional LWI. Conversely, the background model is improved by the joint inversion with the reflectivity in comparison with wave-equation migration velocity analysis. We perform tests on 2D synthetics and 3D field data that demonstrate both benefits.

Deep learning for velocity model building with common-image gather volumes

SUMMARY Subsurface velocity model building is a crucial step for seismic imaging. It is a challenging problem for conventional methods such as full-waveform inversion (FWI) and wave equation migration velocity analysis (WEMVA), due to the highly nonlinear relationship between subsurface velocity values and seismic responses. In addition, traditional FWI and WEMVA methods are often computationally expensive. In this paper, we propose to apply a deep learning technique to construct subsurface velocity models automatically from common-image gather (CIG) volumes. In our method, pairs of synthetic velocity models and CIG volumes are generated to train a convolutional neural network. Our proposed network achieves promising results on different synthetic data sets. The training performance of several commonly used loss functions is also studied.

Differential semblance optimisation based on the adaptive quadratic Wasserstein distance

Abstract As the robustness for the wave equation-based inversion methods, wave equation migration velocity analysis (WEMVA) is stable for overcoming the multipathing problem and has become popular in recent years. As a rapidly developed method, differential semblance optimisation (DSO) is convenient to implement and can automatically detect the moveout existing in common image gathers (CIGs). However, by implementing in the image domain with the target of minimising moveouts and improving coherence of the CIGs, the DSO method often suffers from imaging artefacts caused by uneven illumination and irregular observation geometry, which may produce poor velocity updates with artefact contamination. To deal with this issue, in this paper, by introducing Wiener-like filters, we modify the conventional image matching-based objective function to a new one by introducing the quadratic Wasserstein metric technique. The new misfit function measures the distance of two distributions obtained by the convolutional filters and target functions. With the new misfit function, the adjoint sources and the corresponding gradients are improved. We apply the new method to two numerical examples and one field dataset. The corresponding results indicate that the new method is robust to compensate low frequency components of velocity models.

Adjoint-Driven Deep-Learning Seismic Full-Waveform Inversion

Seismic full-waveform inversion (FWI) aims to build high-resolution images of the physical properties of the subsurface. However, the ill-posedness and nonlinear problems pose a great challenge to the high-resolution reconstruction. Although the nonlinear problem can be mitigated by matching a subset of observation data, the resulting images are generally low-resolution background structures. Regularization-based techniques can mitigate the ill-posedness of FWI, but the iterative method suffers from the cycle-skipping and computational burden problems. To overcome these problems, we develop an adjoint-driven deep-learning FWI (AD-DLFWI) approach which utilizes the fully convolutional network (FCN) to invert subsurface velocity from reflection seismic data. Specifically, AD-DLFWI is implemented in a two-step iterative scheme, in which an optimal update result at each step is learned via a FCN-based learned updating operator. The proposed approach uses the seismic image of applying the adjoint operator of the scattering wave equation, which is equivalent to the gradient of classical FWI, as the data engine of FCN. Inspired by the wave-equation migration velocity analysis approach, we propose to unfold the gradient of FWI into the common-source domain to keep the information about the measure of velocity error. To ensure the interpretability of each network’s role, we design a two-step training scheme to successively reconstruct the low and high wavenumber components of subsurface velocity. Using synthetic experiments with reflection-dominant seismic data, we have confirmed that the proposed FWI approach not only can provide a reliable velocity estimation but also is not sensitive to the cycle-skipping problem.

Wave-equation migration velocity analysis in the common-offset domain to avoid cycle-skipping for FWI

Full waveform inversion (FWI) has been used successfully to build high-resolution acoustic velocity models for imaging but can fail when the starting model is far from accurate because the absence of low frequency information can lead to conversion to local minima. Wave-equation based migration velocity analysis (WEMVA), is an iterative method that has successfully been used to provide low-frequency starting models for full waveform inversion (FWI) and avoid cycle-skipping. In this work, we extend WEMVA to the common-offset domain by minimizing an image residual in the common-offset domain, which can be formed after either reverse-time migration (RTM) or least-squares migration (LSM). Although very expensive in comparison to RTM, LSM has an increased amplitude fidelity that helps to ensure that the image residual used in WEMVA is largely a function of moveout instead of errors in amplitude, thus increasing the likelihood of converging to a global minimum. We demonstrate the method, theory and results on three synthetic models of increasing complexity.

Elastic wave-equation migration velocity analysis preconditioned through mode decoupling

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

Automatic wave-equation migration velocity analysis by focusing subsurface virtual sources

Macro-velocity model building is important for subsequent prestack depth migration and full-waveform inversion. Wave-equation migration velocity analysis uses the band-limited waveform to invert for velocity. Normally, inversion would be implemented by focusing the subsurface offset common-image gathers. We reexamine this concept with a different perspective: In the subsurface offset domain, using extended Born modeling, the recorded data can be considered as invariant with respect to the perturbation of the position of the virtual sources and velocity at the same time. A linear system connecting the perturbation of the position of those virtual sources and velocity is derived and solved subsequently by the conjugate gradient method. In theory, the perturbation of the position of the virtual sources is given by the Rytov approximation. Thus, compared with the Born approximation, it relaxes the dependency on amplitude and makes the proposed method more applicable for real data. We determined the effectiveness of the approach by applying the proposed method on isotropic and anisotropic vertical transverse isotropic synthetic data. A real data set example verifies the robustness of the proposed method.

Q-model building using one-way wave-equation migration Q analysis — Part 2: 3D field-data test

We previously evaluated an inversion-based method, wave-equation migration [Formula: see text] analysis (WEMQA), to estimate the quality factor [Formula: see text] model for seismic attenuation. To demonstrate the feasibility of this method, we applied this method to a 3D seismic data set acquired in the North Sea. Attenuation problems caused by a shallow gas and a shallow channel are observed in this field. We aim to characterize these attenuation anomalies. These attenuation anomalies are correlated with low interval velocities. The provided velocity model does not accurately reflect the low-velocity anomalies. Therefore, we first applied wave-equation migration velocity analysis to update the provided velocity model. The updated velocity shows low-velocity regions around the gas and channel features. The subsurface angle gathers migrated using the updated velocity model are flatter, and the events in the migrated images after velocity updating are more coherent. Then, we applied WEMQA [Formula: see text] to invert for the [Formula: see text] model. The inverted [Formula: see text] model detects the shape and location of the gas and channel. Consequently, the migration with the estimated [Formula: see text] anomalies enhances the damped amplitudes and the frequency content of the migrated events corrects the distorted phase of the migrated events and makes them more coherent.

Wave-equation migration velocity analysis using plane-wave common-image gathers

Wave-equation migration velocity analysis (WEMVA) based on subsurface-offset, angle domain, or time-lag common-image gathers (CIGs) requires significant computational and memory resources because it computes higher dimensional migration images in the extended image domain. To mitigate this problem, we have developed a WEMVA method using plane-wave CIGs. Plane-wave CIGs reduce computational cost and memory storage because they are directly calculated from prestack plane-wave migration and the number of plane waves is often much smaller than the number of shots. In the case of an inaccurate migration velocity, the moveout of plane-wave CIGs is automatically picked by a semblance analysis method, which is then linked to the migration velocity update by a connective function. Numerical tests on two synthetic data sets and a field data set validate the efficiency and effectiveness of this method.

Integrated VTI model building with seismic data, geologic information, and rock-physics modeling — Part 1: Theory and synthetic test

Seismic anisotropy plays an important role in structural imaging and lithologic interpretation. However, anisotropic model building is a challenging underdetermined inverse problem. It is well-understood that single component pressure wave seismic data recorded on the upper surface are insufficient to resolve a unique solution for velocity and anisotropy parameters. To overcome the limitations of seismic data, we have developed an integrated model building scheme based on Bayesian inference to consider seismic data, geologic information, and rock-physics knowledge simultaneously. We have performed the prestack seismic inversion using wave-equation migration velocity analysis (WEMVA) for vertical transverse isotropic (VTI) models. This image-space method enabled automatic geologic interpretation. We have integrated the geologic information as spatial model correlations, applied on each parameter individually. We integrate the rock-physics information as lithologic model correlations, bringing additional information, so that the parameters weakly constrained by seismic are updated as well as the strongly constrained parameters. The constraints provided by the additional information help the inversion converge faster, mitigate the ambiguities among the parameters, and yield VTI models that were consistent with the underlying geologic and lithologic assumptions. We have developed the theoretical framework for the proposed integrated WEMVA for VTI models and determined the added information contained in the regularization terms, especially the rock-physics constraints.

Toward a closed loop from seismic imaging to earth-model building

Velocity-model building is the first task of seismic inversion and the foundation of the subsequent data-processing workflow. When the earth velocity becomes multivalued with respect to the propagating direction of the waves, velocity-model building becomes severely underdetermined and nonunique. The traditional workflow separates velocity-model building from lithologic inversion, which hampers both processing steps. An integrated model-building scheme is demonstrated to simultaneously consider prestack seismic data and its structural and lithologic inversion results from a previous iteration. The prestack seismic inversion is performed using wave-equation migration velocity analysis (WEMVA) for vertical transverse isotropic (VTI) models. To constrain the seismic inversion, the geologic information is integrated as spatial-model correlations, and the rock-physics information as lithologic-model correlations. This feedback step completes the loop from seismic imaging to lithologic-model building, where previous rock-physics estimations and geologic interpretations can be validated further and updated in order to constrain the next WEMVA iteration. Improvements from the integrated inversion scheme are shown on a Gulf of Mexico field data set.

Wave-equation migration velocity analysis for VTI models

Anisotropic models are needed for wave simulation and inversion where a complex geologic environment exists. We extended the theory of wave equation migration velocity analysis to build vertical transverse isotropic models. Because of the ambiguity between depth and [Formula: see text] in the acoustic regime, we assumed [Formula: see text] can be accurately obtained from other sources of information, and inverted for the NMO slowness and the anellipticity parameter [Formula: see text]. We combined the differential semblance optimization objective function with the stacking power maximization to evaluate the focusing of the prestack image in the subsurface-offset domain. To regularize the multiparameter inversion, we built a framework to adapt the geologic and the rock physics information to guide the updates in NMO slowness and [Formula: see text]. This regularization step was crucial to stabilize the inversion and to produce geologically meaningful results. We tested the proposed approach on a synthetic data set and a 2D Gulf of Mexico data set starting with a fairly good initial anisotropic model. The inversion results revealed shallow anomalies collocated in NMO velocity and [Formula: see text] and improved the continuity and the resolution of the final stacked images.

Advances in finite-frequency imaging and inversion — Introduction

While real-life seismic data acquired at finite-frequency bands are inherently highly nonlinear in terms of earth properties, seismic imaging and inversion have historically relied on limiting assumptions, such as linearity, with respect to the subsurface structures and infinite bandwidth data. Over the last two decades, however, seismic exploration has seen unprecedented improvements in acquisition systems, forward-modeling algorithms, and computational power that together finally enable geoscientists to begin tackling the nonlinear, finite-frequency aspects of seismic imaging and inversion in realistic subsurface environments. Presently, these problems are under intense investigation by many scientists in both the exploration and global geophysical communities, labeled under the various methods that fall broadly under seismic imaging, inversion, and tomography. Perhaps the most established of such methods, proposed more than 30 years ago by pioneers such as Tarantola, Lailly, and Bamberger, became known as seismic full-waveform inversion (FWI), where the subsurface parameters are estimated through a data-fitting formalism. In parallel to the classical data-fitting approach used in FWI, other alternative approaches have been proposed to address the finite-frequency and nonlinear nature of seismic data, while aiming at overcoming some of the known limitations of traditional FWI. One such example are the so-called image-domain objective functions, which attempt to retrieve an earth model based on the behavior of the subsurface images: differential semblance optimization, wave-equation migration velocity analysis, and subsurface-angle finite-frequency tomography all fall under this …

Linearized wave-equation migration velocity analysis by image warping

Seismic imaging produces images of contrasts in physical parameters in the subsurface, e.g., velocity or impedance. To build such images, a background model describing the wave kinematics in the earth is necessary. In practice, the structural image and background velocity model are unknown and have to be estimated from the acquired data. Migration velocity analysis deals with estimation of the background model in the framework of seismic migration and relies on two main elements: data redundancy and invariance of the structures with respect to different seismic experiments. Because all the experiments probe the same model, the reflectors must be invariant in suitable domains (e.g., shots or reflection angle); the semblance principle is the tool used to measure the invariance of a set of multiple images. We measure the similarity of the structural features between pairs of single-shot migrated images obtained from adjacent experiments. By using the estimated warping vector field between two migrated images, we construct an image perturbation which describes the difference in reflectivity observed by two shots. We derive an expression for the image perturbation that drives a migration velocity analysis procedure based on a linearization of the wave-equation with respect to the model parameters. Synthetic 2D examples show promising results in retrieving errors in the velocity model. This methodology can be directly applied to 3D.

Anisotropic migration velocity analysis using reverse-time migration

Seismic anisotropy, if not accounted for, can cause significant mispositioning of the reflectors in depth-migrated images. Accounting for anisotropy in depth migration requires velocity analysis tools that can estimate the anisotropic background velocity field. We extended wave equation migration velocity analysis to deal with 2D tilted transverse isotropic media. The velocities were obtained automatically by nonlinear optimization of the focusing and stack power of common-image point gathers constructed using an extended imaging condition. We used the elastic two-way wave equation to reconstruct the wavefields needed for the image and gradient computations. This led to an anisotropic migration velocity analysis algorithm based on reverse-time migration. We illustrated the method with synthetic and field data examples based on marine surface seismic acquisition. The results showed that the method significantly improves the quality of the depth-migrated image. However, as is common in the case of velocity analysis using surface seismic data, the estimation of anisotropic parameters seems to be strongly nonunique.

Polarization-based wave-equation migration velocity analysis in acoustic media

The source extrapolation step in wave-equation prestack reverse-time migration gives wavefield polarization information, which can be used to generate angle-domain common-image gathers (ADCIGs) from seismic reflection data from acoustic media. Concatenation of P-wave polarization segments gives wavefield propagation paths (“wavepaths”), which are similar to the raypaths in ray-based velocity tomography. The ADCIGs provide residual depth moveout (RMO) information, from which a system of linear equations is constructed for tomography to solve for the velocity ratio used for velocity updating. An empirical relation between the RMO data and the velocity ratio updates reduces the amount of computation, and is stabilized by the feedback provided by the iterative loop through prestack migration, to RMO, to velocity update, to prestack migration. Correcting the RMOs to flatten the ADGIGs is the convergence condition. Synthetic data for a layered model with a fault successfully illustrates the method.

Tomographic full-waveform inversion (TFWI) by combining FWI and wave-equation migration velocity analysis

Convergence of full waveform inversion can be improved by extending the velocity model along either the subsurface-offset axis or the time-lag axis. The extension of the velocity model along the time-lag axis enables us to linearly model large time shifts caused by velocity perturbations. The extension is based on a new linearization of the scalar wave equation where the extended-velocity perturbation is convolved in time with the Laplacian of the background wavefield. This linearization is accurate for both reflected events and transmitted events and, in particular, for diving waves recorded at large offsets. The modeling capabilities of the proposed linearization enable the simultaneous inversion of reflections and diving waves even when the starting velocity model is far from being accurate. We solve the resulting optimization problem with a nested algorithm. The inner iterations are based on the proposed linearization and on a mixing of scales between the short- and long-wavelength components of the velocity model. Numerical tests performed on two synthetic data sets modeled on the Marmousi model and on the Caspian Sea portion of the well-known BP model show the global-convergence properties as well as the high-resolution potential of the method.