Differential Semblance Optimization Research Articles

Investigating Hessian-based inversion velocity analysis

Inversion velocity analysis (IVA) is an image-domain method built upon the spatial scale separation of the model. Accordingly, the IVA method is performed with an iterative process composed of two minimization steps consisting of migration (inner loop) and tomography (outer loop), respectively, with each step accounting for its Hessian or not. The migration part provides the common-image gathers (CIGs) with extension in the horizontal subsurface offset. Then, the differential semblance optimization (DSO) misfit measures the focusing of the events in the CIGs, which indicates the quality of the velocity model. Commonly, the velocity updates are obtained from the DSO gradient. IVA is a modified version in which the approximate inverse replaces the adjoint of the inner loop process; in that case, the migration Hessian is approximately diagonal in the high-frequency regime. In this work, we report on implementation of the tomographic Hessian (i.e., the second derivative of the DSO misfit with respect to the background model) for the estimation of the background velocity model. We apply the second-order adjoint-state method to obtain the application of the tomographic Hessian on a vector. Then, we use the truncated-Newton (TN) method to obtain the update directions by computing approximately the application of the inverse of the tomographic Hessian on the descent direction. We also make a theoretical comparison between tomography in the IVA and full-waveform inversion contexts. Two numerical examples are used to compare, in terms of geophysical results and computational costs, the TN method with different gradient-based optimization methods applied to the IVA. A small model allows us to evaluate the eigenvalues of the tomographic Hessian, which explains the large damping needed in the TN case.

Seismic differential semblance-oriented migration velocity analysis — Status and the way forward

Differential semblance optimization (DSO) is a seismic waveform inversion approach. It has been developed to remove the limitation of conventional full-waveform inversion as a velocity model building tool when only reflected waves are available. It has connections with migration velocity analysis (MVA) techniques defined in the image domain to build macrovelocity models, with a split between the macromodel and the small-scale heterogeneities. Among the various DSO approaches, DSO-MVA has unique characteristics for velocity model building of complex structures, while ensuring the convergence to the global minimum in a local optimization approach: the wavefields are computed with the two-way wave equation operators, focusing panels are built in a survey-sinking mode, and the quality of the macrovelocity model is evaluated on common-image gathers in the subsurface-offset domain through a differential annihilator. We review the significant aspects that have been better understood over the past 15 years and discuss the way forward. The first identified issue has been the imprint of the small-scale heterogeneities on the macromodel update that is revealed with the use of wave equation-based Green’s functions and an ad hoc solution is subsequently developed by slightly modifying the definition of the objective function. A second important aspect has been the development of efficient approaches to compute the required quantitative migration based on approximate inverses to replace the previously used migrations. The issue of correctly interpreting the amplitudes has also moved into one of an improved model parameterization, i.e., from constant-density acoustics media to full acoustic media as a path toward fully viscoelastic media. Finally, we discuss the way forward with the challenges of extending DSO-MVA beyond the single scattering approximation, for processing diving waves and multiples, and also the challenge of numerical implementation, which remains critical even in two dimensions, for applicability to large-scale real data.

A new data-domain reflection full-waveform inversion method based on common-image-point gathers

The migration velocity analysis (MVA) based on the common-image gathers (CIGs) in the image domain is an effective approach for evaluating the seismic background velocity, a crucial index for the full-waveform inversion (FWI) and migration. Compared with cumbersome conventional MVA methods, differential semblance optimization (DSO) is a variation of image-domain wavefield tomography that avoids interactive manual picking but increases the computation cost tremendously. In recent years, the reflection waveform inversions have been studied in depth to construct the background velocity, matching reflection travel time or waveform in the data domain, which requires expensive Born simulations based on accurate imaging results. This study performs a new reflection full-waveform inversion method in the image domain based on the flatness of arbitrarily selected common-image-point (CIP) gathers and their Born simulations. According to the relationship between imaging and background model, the flatness of CIP gathers under the L1 norm is measured. Afterwards, we construct the objective function to update the background model without manual picking or wave-equation migration in an extended dimension. The inversion accuracy is controlled by balancing the density of the imaging points and the computation capability. The whole procedure is implemented in the frequency domain, so the main computation comes from the LU decomposition in the frequency-domain wave-equation simulation, in which way, the proposed data-domain reflection full-waveform inversion achieves complete automation with lower costs. The objective function analysis shows a better convex, proving the weak dependence on the starting model of this method. Besides, synthetic experiments verify the validity and the field data test demonstrates its effectiveness.

A robust migration velocity analysis method based on adaptive differential semblance optimization

Migration velocity analysis (MVA) is an efficient tool to reconstruct low wavenumber components of the model. Compared with data domain methods, the MVA methods are implemented in image domain by processing imaging results directly through minimizing moveouts and improving the coherence of the common image gathers (CIGs), such as angle domain common image gathers (ADCIGs) or subsurface offset domain image gathers (ODCIGs). As one of the wave-equation based migration velocity analysis (WEMVA) methods, differential semblance optimization (DSO) can automatically detect the moveout existed in the CIGs and is convenient to implement. However, referring to the CIGs contaminated by spurious imaging artefacts caused by uneven illumination and irregular observation geometry, the traditional DSO method may produce poor velocity update with oscillations, which can prevent rapid convergence to a correct velocity model. To deal with this issue and extend the solution space, we introduce an adaptive DSO (ADSO) method by replacing the traditional image matching with energy distribution matching. By reducing dependence on the imaging results and introducing a variable weighting function in the calculation of the adjoint sources, the amplitude of the inverted model can be evenly distributed and free of artefacts. Three numerical examples show that the ADSO method is robust with little artefacts presenting in the gradients. Combined with the improved gradient, the ADSO can lead to a stable update.

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.

Passive-seismic image-domain elastic wavefield tomography

SUMMARY Elastic time-reverse imaging offers a robust wavefield-based approach for locating microseismic events; however, event location accuracy greatly depends on the veracity of the elastic velocity models (i.e. VP and VS) used for wave propagation. In this study, we propose a methodology for microseismic image-domain wavefield tomography using the elastic wave equation and zero-lag and extended source images, the focusing of which is used as a quality control metric for velocity models. The objective function is designed to measure the focusing of time-reversed microseismic energy in zero-lag and extended event images. The function applies penalty operators to source images to highlight poorly focused residual energy caused by backpropagation through erroneous velocity models. Minimizing the objective function leads to a model optimization problem aimed at improving the image-focusing quality. P- and S-wave velocity model updates are computed using the adjoint-state method and build on the zero-lag and extended image residuals that satisfy the differential semblance optimization criterion. Synthetic experiments demonstrate that one can construct accurate elastic velocity models using the proposed method, which can significantly improve the focusing of imaged events leading to, for example, enhanced fluid-injection programs.

The effectiveness of a pseudo‐inverse extended Born operator to handle lateral heterogeneity for imaging and velocity analysis applications

ABSTRACTWave equation–based migration velocity analysis techniques aim to construct a kinematically accurate velocity model for imaging or as an initial model for full waveform inversion applications. The most popular wave equation–based migration velocity analysis method is differential semblance optimization, where the velocity model is iteratively updated by minimizing the unfocused energy in an extended image volume. However, differential semblance optimization suffers from artefacts, courtesy of the adjoint operator used in imaging, leading to poor convergence. Recent findings show that true amplitude imaging plays a significant role in enhancing the differential semblance optimization's gradient and reducing the artefacts. Here, we focus on a pseudo‐inverse operator to the horizontally extended Born as a true amplitude imaging operator. For laterally inhomogeneous models, the operator required a derivative with respect to a vertical shift. Extending the image vertically to evaluate such a derivative is costly and impractical. The inverse operator can be simplified in laterally homogeneous models. We derive an extension of the approach to apply the full inverse formula and evaluate the derivative efficiently. We simplified the implementation by applying the derivative to the imaging condition and utilize the relationship between the source and receiver wavefields and the vertical shift. Specifically, we verify the effectiveness of the approach using the Marmousi model and show that the term required for the lateral inhomogeneity treatment has a relatively small impact on the results for many cases. We then apply the operator in differential semblance optimization and invert for an accurate macro‐velocity model, which can serve as an initial velocity model for full waveform inversion.

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.

Coupling direct inversion to common-shot image-domain velocity analysis

Migration velocity analysis is a technique used to estimate the large-scale structure of the subsurface velocity model controlling the kinematics of wave propagation. For more stable results, recent studies have proposed to replace migration, adjoint of Born modeling, by the direct inverse of the modeling operator in the context of extended subsurface-offset domain. Following the same strategy, we have developed a two-way-wave-equation-based inversion velocity analysis (IVA) approach for the original surface-oriented shot gathers. We use the differential semblance optimization (DSO) objective function to evaluate the quality of inverted images depending on shot positions and to derive the associated gradient, an essential element to update the macromodel. We evaluate the advantages and limitations through applications of 2D synthetic data sets, first on simple models with a single-reflector embedded in various background velocities and then on the Marmousi model. The direct inverse attenuates migration smiles by compensating for geometric spreading and uneven illuminations. We slightly modified the original DSO objective function to remove spurious oscillations around interface positions in the velocity gradient. These oscillations are related to the fact that the locations of events in the image domain depend on the macromodel. We pay attention to the presence of triplicated wavefields. It appears that IVA is robust even if artifacts are observed in the seismic migrated section. The velocity gradient leads to a stable update, especially after a Gaussian smoothing over a wavelength distance. Coupling common-shot direct inversion to velocity analysis offers new possibilities for the extension to 3D in the future.

Toward a stable iterative migration velocity analysis scheme

Migration velocity analysis is a family of methods aiming at automatically recovering large-scale trends of the velocity model from primary reflection data. We studied an image-domain version, in which the model is extended with the subsurface offset and we use the differential semblance optimization objective function. To incorporate first-order surface multiples in this method, the standard migration step is replaced by a least-squares iterative scheme aiming at determining an extended reflectivity model explaining primaries and multiples. Hence, this iterative migration velocity analysis strategy takes the form of a nested optimization problem, with gradient-based minimization techniques for the inner (migration part) and outer loops (macromodel estimation). The behavior of the outer loop gradient is unstable, depending on the number of iterations of the inner loop. This problem is addressed by slightly modifying the outer loop objective function: A “filter” operator attenuating unwanted energy in the extended reflectivity is applied before evaluating the focusing of reflectivity images. Simple synthetic numerical examples illustrate that this modification improves the stability of the gradient. In addition, a less expensive outer gradient computation is proposed, without harming the background velocity updates.

Inversion velocity analysis in the subsurface-offset domain

Optimization-based migration velocity analysis updates long-wavelength velocity information by minimizing an objective function that measures the violation of a focusing criterion, applied to an image volume. Differential semblance optimization forms a smooth objective function in velocity and data, regardless of the data-frequency content. Depending on how the image volume is formed, however, the objective function may not be minimized at a kinematically correct velocity, a phenomenon characterized in the literature (somewhat inaccurately) as “gradient artifacts.” We find that the root of this pathology is imperfect image volume formation resulting from reverse time migration (RTM), and that the use of linearized inversion (least-squares migration) more or less eliminates it. A synthetic Marmousi example and a 2D real data example are used to demonstrate that an approximate inverse operator, a little more expensive than RTM, leads to recovery of a kinematically correct velocity.

Inversion gradients for acoustic VTI wavefield tomography

Wavefield tomography can handle complex subsurface geology better than ray-based techniques and, ultimately, provide a higher resolution. Here, we implement forward and adjoint wavefield extrapolation for VTI (transversely isotropic with a vertical symmetry axis) media using a generalized pseudospectral operator based on a separable approximation for the P-wave dispersion relation. This operator is employed to derive the gradients of the differential semblance optimization (DSO) and modified image-power objective functions. We also obtain the gradient expressions for a data-domain objective function that can more easily incorporate borehole information necessary for stable VTI velocity analysis. These gradients are similar to the ones obtained with a space-time finite-difference (FD) scheme for a system of coupled wave equations but the pseudospectral method is not hampered by the imprint of the shear-wave artifact. Numerical examples also show the potential advantages of the modified image-power objective function in estimating the anellipticity parameter [Formula: see text].

Preconditioned prestack plane-wave least squares reverse time migration with singular spectrum constraint

Least squares migration can eliminate the artifacts introduced by the direct imaging of irregular seismic data but is computationally costly and of slow convergence. In order to suppress the migration noise, we propose the preconditioned prestack plane-wave least squares reverse time migration (PLSRTM) method with singular spectrum constraint. Singular spectrum analysis (SSA) is used in the preconditioning of the take-offangle-domain common-image gathers (TADCIGs). In addition, we adopt randomized singular value decomposition (RSVD) to calculate the singular values. RSVD reduces the computational cost of SSA by replacing the singular value decomposition (SVD) of one large matrix with the SVD of two small matrices. We incorporate a regularization term into the preconditioned PLSRTM method that penalizes misfits between the migration images from the plane waves with adjacent angles to reduce the migration noise because the stacking of the migration results cannot effectively suppress the migration noise when the migration velocity contains errors. The regularization imposes smoothness constraints on the TADCIGs that favor differential semblance optimization constraints. Numerical analysis of synthetic data using the Marmousi model suggests that the proposed method can efficiently suppress the artifacts introduced by plane-wave gathers or irregular seismic data and improve the imaging quality of PLSRTM. Furthermore, it produces better images with less noise and more continuous structures even for inaccurate migration velocities.

Analysis of RTM extended images for VTI media

Extended images obtained from reverse time migration (RTM) contain information about the accuracy of the velocity field and subsurface illumination at different incidence angles. Here, we evaluate the influence of errors in the anisotropy parameters on the shape of the residual moveout (RMO) in P-wave RTM extended images for VTI (transversely isotropic with a vertical symmetry axis) media. Using the actual spatial distribution of the zero-dip NMO velocity ([Formula: see text]), which could be approximately estimated by conventional techniques, we analyze the extended images obtained with distorted fields of the parameters [Formula: see text] and [Formula: see text]. Differential semblance optimization (DSO) and stack-power estimates are employed to study the sensitivity of focusing to the anisotropy parameters. We also build angle gathers to facilitate interpretation of the shape of RMO in the extended images. The results show that the signature of [Formula: see text] is dip-dependent, whereas errors in [Formula: see text] cause defocusing only if that parameter is laterally varying. Hence, earlier results regarding the influence of [Formula: see text] and [Formula: see text] on reflection moveout and migration velocity analysis remain generally valid in the extended image space for complex media. The dependence of RMO on errors in the anisotropy parameters provides essential insights for anisotropic wavefield tomography using extended images.

Image‐warping waveform tomography

ABSTRACTImaging the change in physical parameters in the subsurface requires an estimate of the long wavelength components of the same parameters in order to reconstruct the kinematics of the waves propagating in the subsurface. One can reconstruct the model by matching the recorded data with modeled waveforms extrapolated in a trial model of the medium. Alternatively, assuming a trial model, one can obtain a set of images of the reflectors from a number of seismic experiments and match the locations of the imaged interfaces. Apparent displacements between migrated images contain information about the velocity model and can be used for velocity analysis. A number of methods are available to characterize the displacement between images; in this paper, we compare shot‐domain differential semblance (image difference), penalized local correlations, and image‐warping. We show that the image‐warping vector field is a more reliable tool for estimating displacements between migrated images and leads to a more robust velocity analysis procedure. By using image‐warping, we can redefine the differential semblance optimization problem with an objective function that is more robust against cycle‐skipping than the direct image difference. We propose an approach that has straightforward implementation and reduced computational cost compared with the conventional adjoint‐state method calculations. We also discuss the weakness of migration velocity analysis in the migrated‐shot domain in the case of highly refractive media, when the Born modelling operator is far from being unitary and thus its adjoint (migration) operator poorly approximates the inverse.

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 …

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.

Subsalt velocity estimation by target-oriented wave-equation migration velocity analysis: A 3D field-data example

We apply target-oriented wave-equation migration velocity analysis to a 3D field data set acquired from the Gulf of Mexico. Instead of using the original surface-recorded data set, we use a new data set synthesized specifically for velocity analysis to update subsalt velocities. The new data set is generated based on an initial unfocused target image and by a novel application of 3D generalized Born wavefield modeling, which correctly preserves velocity kinematics by modeling zero and nonzero subsurface-offset-domain images. The target-oriented inversion strategy drastically reduces the data size and the computation domain for 3D wave-equation migration velocity analysis, greatly improving its efficiency and flexibility. We apply differential semblance optimization (DSO) using the synthesized new data set to optimize subsalt velocities. The updated velocity model significantly improves the continuity of subsalt reflectors and yields flattened angle-domain common-image gathers.

Automatic velocity analysis by differential semblance optimization

We present a method for automatic velocity analysis of seismic data based on differential semblance optimization (DSO). The data are mapped for each offset from the time domain to the depth domain by a Born migration scheme using ray tracing with the efficient wavefront construction method. The DSO cost functional is evaluated by taking differences of the migration images for neighboring offsets. The gradient of this functional with respect to the underlying velocity model is obtained by a first‐order approximation of the adjoint‐state method, leading to an optimal complexity: the cost of evaluating the gradient is about the same as that of evaluating the functional. The method has been applied to a marine line. Multiples turned out to be a problem, but were handled effectively by incorporating a multiple filter inside the DSO cost functional.

Two-dimensional velocity macro model estimation from seismic reflection data by local differential semblance optimization: applications to synthetic and real data sets

SUMMARY The quality of the migration/inversion in seismic reflection is directly related to the quality of the velocity macro model. We present here an extension of the differential semblance optimization method (DSO) for 2-D velocity field estimation. DSO evaluates via local measurements (horizontal derivatives) how flat events in common-image gathers are. Its major advantage with respect to the usual cost functions used in reflection seismic inverse problems is that it is—at least in the 1-D case—unimodal and thus allows a local (gradient) optimization process. Extension of DSO to three dimensions in real cases involving a large number of inverted parameters thus appears much more feasible, because convergence might not require a random search process (global optimization). Our differential semblance function directly measures the quality of the commonimage gathers in the depth-migrated domain and does not involve de-migration. An example of inversion on a 2-D synthetic data set shows the ability of DSO to handle 2-D media with local optimization algorithms. The horizontal derivatives have to be carefully calculated for the inversion process. However, the computation of only a few commonimage gathers is sufficient for a stable inversion. As a Kirchhoff scheme is used for migration, this undersampling largely reduces the computational cost. Finally, we present an application to a real North Sea marine data set. We prove with this example that DSO can provide velocity models for typical 2-D acquisition that improve the quality of the final pre-stack depth images when compared to the quality of images migrated with a velocity model obtained by a classical NMO/DMO analysis. Whilst random noise is not a real difficulty for DSO, coherent noise, however, has to be carefully eliminated before or during inversion for the success of the velocity estimation.