Velocity analysis for plane‐wave source migration using the finite‐frequency sensitivity kernel

The idea of path-integral imaging is to sum over the migrated images obtained for a set of migration velocity models. Velocities where common-image gathers align horizontally are stationary, thus favoring these images in the overall stack. The overall image forms with no knowledge of the true velocity model. However, the velocity information associated with the final image can be determined in the process. By executing the path-integral imaging twice and weighting one of the stacks with the velocity value, the stationary velocities that produce the final image can then be extracted by a division of the two images. The velocity extraction, interpolation, and smoothing can be done fully automatically, without the need for human interpretation or other intervention. A numerical example demonstrated that quantitative information about the migration velocity model can be determined by double path-integral migration. The so-obtained velocity model can then be used as a starting model for subsequent velocity analysis tools like migration velocity analysis or tomographic methods.

ABSTRACTLeast‐squares inversion of seismic reflection waveform data can reconstruct remarkably detailed models of subsurface structure and take into account essentially any physics of seismic wave propagation that can be modelled. However, the waveform inversion objective has many spurious local minima, hence convergence of descent methods (mandatory because of problem size) to useful Earth models requires accurate initial estimates of long‐scale velocity structure. Migration velocity analysis, on the other hand, is capable of correcting substantially erroneous initial estimates of velocity at long scales. Migration velocity analysis is based on prestack depth migration, which is in turn based on linearized acoustic modelling (Born or single‐scattering approximation). Two major variants of prestack depth migration, using binning of surface data and Claerbout's survey‐sinking concept respectively, are in widespread use. Each type of prestack migration produces an image volume depending on redundant parameters and supplies a condition on the image volume, which expresses consistency between data and velocity model and is hence a basis for velocity analysis. The survey‐sinking (depth‐oriented) approach to prestack migration is less subject to kinematic artefacts than is the binning‐based (surface‐oriented) approach. Because kinematic artefacts strongly violate the consistency or semblance conditions, this observation suggests that velocity analysis based on depth‐oriented prestack migration may be more appropriate in kinematically complex areas. Appropriate choice of objective (differential semblance) turns either form of migration velocity analysis into an optimization problem, for which Newton‐like methods exhibit little tendency to stagnate at nonglobal minima. The extended modelling concept links migration velocity analysis to the apparently unrelated waveform inversion approach to estimation of Earth structure: from this point of view, migration velocity analysis is a solution method for the linearized waveform inversion problem. Extended modelling also provides a basis for a nonlinear generalization of migration velocity analysis. Preliminary numerical evidence suggests a new approach to nonlinear waveform inversion, which may combine the global convergence of velocity analysis with the physical fidelity of model‐based data fitting.

Reverse time migration (RTM) is achieved by a forward and reverse time propagation of source and receiver wavefields respectively, followed by an imaging condition. The quality of the image from any RTM implementation depends directly on the method’s ability to separate the true reflection data from the backscattered correlation noise between the source and receiver wavefields. In this paper we demonstrate the application of an inverse scattering imaging condition that significantly reduces this backscattered correlation noise and results in a much higher quality subsurface image than is achieved by typical RTM imaging conditions.

An integrated procedure for migration velocity analysis in complex structures of thrust belts

This research evaluated team members’ perceptions of alternative decision analysis approaches when they were asked to analyze a simple or complex data set under a time constraint. There has been significant work done to understand what criteria should be included in this class of problems. However, less work has been done examining the selection of a specific decision analysis approach. This is an area of concern since research has shown that selecting a different decision analysis approach can change the results even when the same data is analyzed. This research effort is a preliminary look at perceptions of alternative decision analysis approaches. A mixed method study analyzing quantitative and qualitative data was completed. Teams were assigned to use one of five different decision analysis approaches, including Weighted Sum, SMART, TOPSIS, AHP, and ELECTRE I. After applying the approach to either a simple or complex data set, team members were surveyed to determine perceptions including comfort with the approach, its results, and ease/difficulty of explaining the approach to coworkers. Discussions were also held with team members to gain additional insights. The study indicated that team members preferred less technically challenging decision analysis approaches, such as Weighted Sum and to a lesser degree SMART, when they were using a simple data set because they found the approach easier to use and to explain to others. The more technically challenging decision analysis approaches, such as AHP and ELECTRE I, were preferred with a complex data set because they provided better insight into the problem. Teams in this study did selected different alternatives when they used different decision analysis approaches. Demographic data was also collected but was not found to influence results in this study.

Summary Seismic exploration of geothermal resources in the German Molasse Basin (GMB) requires a proper seismic imaging of deep productive horizons. Success of this process is heavily reliant on the correct estimation of seismic velocities by migration velocity analysis (MVA). However, the MVA has a high computational cost and, depending on the quality of seismic data and approach chosen for the analysis, it might produce erroneous results. In this work, we developed an MVA workflow that enables to create a velocity model that is consistent with seismic, geological and borehole data, despite the limitations inherent to the MVA. A 2D seismic data set acquired in the GMB was used to test the MVA. The results were evaluated based on four criteria: increase in seismic resolution, availability of good quality reflectors beneath velocity components, geological plausibility and consistency with borehole data. Such an interpretive approach to the initially purely processing procedure allowed to discriminate genuine velocity components from erroneous components. Our work shows that the MVA produces an accurate velocity model of the subsurface, if the data have good quality, sufficient cable length, and geological constraints in form of lithological and velocity logs from boreholes in the vicinity of a study area.

Not giving mathematical details but the simple principles, we present two types of techniques for both depth and time prestack migration velocity analysis. The first type of the techniques is based on the migration velocity spectrum, and the second is based on the migration velocity scanning. The accuracy of the picked/selected velocities can be tested by the common image gathers with a common-shot style. An alternate approach, based on the so-called zero-timedifference gathers computed by reflection or scattering ray tracing regarding to the migrated horizon(s), is proposed to be used as an additional way for migration velocity testing in complex areas.

The idea of path-integral imaging is to sum over the migrated images obtained for a set of migration velocity models. Those velocities where common-image gathers align horizontally are stationary, thus favoring these images in the overall stack. In this way, the overall image forms with no knowledge of the true velocity model. However, the velocity information associated with the final image can be determined in the process. By executing the path-integral imaging twice, weighting one of the stacks with the velocity value, the stationary velocities that produce the final image can then be extracted by a division of the two images. A numerical example demonstrates that quantitative information about the migration velocity model can be determined by double path-integral migration.

Low relief structures are increasingly important onshore exploration targets. Accurate resolution of these structures and reservoir characterization can be difficult because seismic reflections of these targets are contaminated with distortions due to complicated near surface geology. Downward continuation wave equation datuming techniques offer a solution to the near surface problem but they require a detailed knowledge of the near surface velocity. Near surface Migration Velocity Analysis (MVA) can be quite challenging due to the low fold of the data involved. In this paper, we integrate different MVA methods into the common image cube (CIC) approach. These methods include residual curvature analysis (RCA), depth focusing analysis, and common focus point (CFP). We use the CIC as a domain for near surface velocity analysis to address the low fold problem. For 2D data at a lateral position, instead of invoking the zero-lag imaging condition, we store other lags at each depth level. The result is a cube of data that offers more information than other methods. We illustrate two uses of this cube by applying wave equation datuming to a synthetic dataset, simulated from topography, using correct and incorrect velocity models, respectively.

ABSTRACTWe present a migration velocity analysis (MVA) method based on wavefield extrapolation. Similarly to conventional MVA, our method aims at iteratively improving the quality of the migrated image, as measured by the flatness of angle‐domain common‐image gathers (ADCIGs) over the aperture‐angle axis. However, instead of inverting the depth errors measured in ADCIGs using ray‐based tomography, we invert ‘image perturbations’ using a linearized wave‐equation operator. This operator relates perturbations of the migrated image to perturbations of the migration velocity. We use prestack Stolt residual migration to define the image perturbations that maximize the focusing and flatness of ADCIGs.Our linearized operator relates slowness perturbations to image perturbations, based on a truncation of the Born scattering series to the first‐order term. To avoid divergence of the inversion procedure when the velocity perturbations are too large for Born linearization of the wave equation, we do not invert directly the image perturbations obtained by residual migration, but a linearized version of the image perturbations. The linearized image perturbations are computed by a linearized prestack residual migration operator applied to the background image. We use numerical examples to illustrate how the backprojection of the linearized image perturbations, i.e. the gradient of our objective function, is well behaved, even in cases when backprojection of the original image perturbations would mislead the inversion and take it in the wrong direction.We demonstrate with simple synthetic examples that our method converges even when the initial velocity model is far from correct. In a companion paper, we illustrate the full potential of our method for estimating velocity anomalies under complex salt bodies.

ABSTRACTAngle‐domain common‐image gathers (ADCIGs) are an essential tool for migration velocity analysis (MVA). We present a method for computing ADCIGs in 3D from the results of wavefield‐continuation migration. The proposed methodology can be applied before or after the imaging step in a migration procedure. When computed before imaging, 3D ADCIGs are functions of the offset ray parameters (p, p); we derive the geometric relationship that links the offset ray parameters to the aperture angle γ and the reflection azimuth φ. When computed after imaging, 3D ADCIGs are directly produced as functions of γ and φ.The mapping of the offset ray parameters (p, p) into the angles (γ, φ) depends on both the local dips and the local interval velocity; therefore, the transformation of ADCIGs computed before imaging into ADCIGs that are functions of the actual angles is difficult in complex structure. By contrast, the computation of ADCIGs after imaging is efficient and accurate even in the presence of complex structure and a heterogeneous velocity function. On the other hand, the estimation of the offset ray parameters (p, p) is less sensitive to velocity errors than the estimation of the angles (γ, φ). When ADCIGs that are functions of the offset ray parameters (p, p) are adequate for the application of interest (e.g. ray‐based tomography), the computation of ADCIGs before imaging might be preferable.Errors in the migration velocity cause the image point in the angle domain to shift along the normal to the apparent geological dip. By assuming stationary rays (i.e. small velocity errors), we derive a quantitative relationship between this normal shift and the traveltime perturbation caused by velocity errors. This relationship can be directly used in an MVA procedure to invert depth errors measured from ADCIGs into migration velocity updates. In this paper, we use it to derive an approximate 3D residual moveout (RMO) function for measuring inconsistencies between the migrated images at different γ and φ. We tested the accuracy of our kinematic analysis on a 3D synthetic data set with steeply dipping reflectors and a vertically varying propagation velocity. The tests confirm the accuracy of our analysis and illustrate the limitations of the straight‐ray approximation underlying our derivation of the 3D RMO function.

ABSTRACTWave‐equation migration velocity analysis is a technique designed to extract and update velocity information from migrated images. The velocity model is updated through the process of optimizing the coherence of images migrated with the known background velocity model. The capacity for handling multi‐pathing of the technique makes it appropriate in complex subsurface regions characterized by strong velocity variation. Wave‐equation migration velocity analysis operates by establishing a linear relation between a slowness perturbation and a corresponding image perturbation. The linear relationship and the corresponding linearized operator are derived from conventional extrapolation operators and the linearized operator inherits the main properties of frequency‐domain wavefield extrapolation. A key step in the implementation is to design an appropriate procedure for constructing an image perturbation relative to a reference image that represents the difference between the current image and a true, or more correct image of the subsurface geology. The target of the inversion is to minimize such an image perturbation by optimizing the velocity model. Using time‐shift common‐image gathers, one can characterize the imperfections of migrated images by defining the focusing error as the shift of the focus of reflections along the time‐shift axis. The focusing error is then transformed into an image perturbation by focusing analysis under the linear approximation. As the focusing error is caused by the incorrect velocity model, the resulting image perturbation can be considered as a mapping of the velocity model error in the image space. Such an approach for constructing the image perturbation is computationally efficient and simple to implement. The technique also provides a new alternative for using focusing information in wavefield‐based velocity model building. Synthetic examples demonstrate the successful application of our method to a layered model and a subsalt velocity update problem.

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.

Migration velocity analysis (MVA) is a technique defined in the image domain to determine the background velocity model controlling the kinematics of wave propagation. In the presence of discontinuous interfaces, the velocity gradient used to iteratively update the velocity model exhibits spurious oscillations. For more stable results, we replace the migration part by an inversion scheme. By definition, migration is the adjoint of the Born modeling operator, whereas inversion is its asymptotic inverse. We have developed new expressions in 1D and 2D cases based on two-way wave-equation operators. The objective function measures the quality of the images obtained by inversion in the extended domain depending on the subsurface offset. In terms of implementation, the new approach is very similar to classic MVA. A 1D analysis found that oscillatory terms around the interface positions can be removed by multiplying the inversion result with the velocity at a specific power before evaluating the objective function. Several 2D synthetic data sets are discussed through the computation of the gradient needed to update the model parameters. Even for discontinuous reflectivity models, the new approach provides results without artificial oscillations. The model update corresponds to a gradient of an existing objective function, which was not the case for the horizontal contraction approach proposed as an alternative to deal with gradient artifacts. It also correctly handles low-velocity anomalies, contrary to the horizontal contraction approach. Inversion velocity analysis offers new perspectives for the applicability of image-domain velocity analysis.

The determination of topographical macro‐velocity model is of great significance for seismic imaging in the mountainous or piedmont area, such as West and South China. Therefore, an effective procedure for migration velocity analysis on topography is needed. The residual curvature analysis (RCA) based on the CIGs is an effective method to improve the accuracy of the topographical macro‐velocity model. Based on the offset CIGs generated by rugged topography Kirchhoff PSDM, we propose a complete depth migration velocity analysis workflow which includes the conversion of average velocity to interval velocity and the updating of rugged topography migration velocity model. Numerical tests on Field data from mountainous area reveal the effectiveness of this workflow.