Migration velocity analysis and waveform inversion

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

One of the most serious problems in anisotropic velocity analysis is the trade-off between anisotropy and lateral heterogeneity, especially if velocity varies on a scale smaller than the maximum offset. We have developed a P-wave MVA (migration velocity analysis) algorithm for transversely isotropic (TI) models that include layers with small-scale lateral heterogeneity. Each layer is described by constant Thomsen parameters e and δ and the symmetry-direction velocity V0 that varies as a quadratic function of the distance along the layer boundaries. For tilted TI media (TTI), the symmetry axis is taken orthogonal to the reflectors. We analyzed the influence of lateral heterogeneity on image gathers obtained after prestack depth migration and found that quadratic lateral velocity variation in the overburden can significantly distort the moveout of the target reflection. Consequently, medium parameters beneath the heterogeneous layer(s) are estimated with substantial error, even when borehole information (e.g., check shots or sonic logs) is available. Because residual moveout in the image gathers is highly sensitive to lateral heterogeneity in the overburden, our algorithm simultaneously inverts for the interval parameters of all layers. Synthetic tests for models with a gently dipping overburden demonstrate that if the vertical profile of the symmetry-direction velocity V0 is known at one location, the algorithm can reconstruct the other relevant parameters of TI models. The proposed approach helps increase the robustness of anisotropic velocity model-building and enhance image quality in the presence of small-scale lateral heterogeneity in the overburden.

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.

The Guando field is in an area of rough topography and complex geology. In order to better site wells in such fields it is of value to have a sound understanding of this geological complexity. Obtaining a satisfactory seismic image to enable such interpretation requires pre-stack imaging. The geological complexity is such that post stack imaging fails due to the failure of the common mid point assumption. Is it sufficient to perform pre-stack time migration or is it necessary to perform pre-stack depth migration? In this instance both have been performed on this field and results for each of them will be shown. The basic theory of both pre-stack time and pre-stack depth migration will be described without using mathematics. Model data will then be used to illustrate the shortcomings of pre-stack time migration using a model of similar geological complexity to the Guando field. Both pre-stack migrations require a velocity field. These differ and the manner of their derivation also differs. Velocity model building for both techniques will be described. Particular emphasis will be placed on the use of tomographic and scanning techniques for pre-stack depth migration. Examples will be shown of the velocity models from both migrations. It will be shown that the pre-stack depth migration velocity model is also a valuable interpretation asset. Data from both migrations will be shown. These will demonstrate the superiority of the pre-stack depth migrated data and the reasons why it is superior.

This study introduces a Common Offset (CO) extension of the Common Reflection Surface (CRS) method to address seismic imaging challenges in complex geological settings and with noisy data. This CO-CRS approach aims to enhance the signal-to-noise ratio and overcome the limitations of conventional preconditioning techniques that rely on accurate parameterization. Building upon established work on zero-offset CRS (ZO-CRS), the CO method generates regularized prestack data suitable for both time- and depth-domain processing by interpolating missing offsets using a local hyperbolic approximation. Ultimately, this study utilizes CO-CRS for enhanced velocity analysis and data preconditioning prior to performing prestack time migration (PSTM). In this study, the CO-CRS is then used for velocity analysis and prestack time migration. The results show that prestack CO-CRS data yield improved time-migrated seismic images, and we suggest extending the application to the depth domain. To achieve a reliable velocity model for imaging, recursive seismic inversion (RSI) is applied to derive the velocity model using the PSTM stack and a velocity interval time, based on CRS semblance velocity analysis. Furthermore, the prestack depth migration (PSDM) is then tested. The depth-imaging results are reliable, and it can be concluded that combining the benefits of the CRS noise-reduction feature with more accurate velocity analysis and prestack migration can provide enhanced capabilities.

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.

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.

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.

In order to compare prestack depth migration algorithms for imaging complex structures, we constructed typical Foothills geological model that included thrust faults and duplex structures and computed the model response (synthetic seismograms) by ray tracing. We investigated the performance of the Kirchhoff and the explicit finitedifference (also called the f-x) algorithms on these synthetic data. For cases where the correct velocity model was used, comparisons showed the f-x method to be more effective and robust in performing prestack depth migration from topography than the Kirchhoff method. In cases, where the velocity model was not assumed to be known, we performed migration/velocity analysis by examining “smiles/frowns” in common image gathers and determined the macro-velocity model by using “focusing analysis”. We proposed a quantitative migration velocity analysis technique based on the curvature of “smiles/frowns”. The characteristics of coincident time of “smiles/frowns” of this new approach could be used as a diagnostic tool for migration/velocity analysis in the future.

SM1.4 SUMMARY In case of severe lateral velocity variations prestack depth migration is not suitable for migration velocity analysis. We therefore propose to substitute prestack depth migration by prestack imaging by coupled linearized inversion (PICLI). Results obtained with the Marmousi model show the improvement offered by this method for migration velocity analysis. PICLI involves a huge amount of computation. Hence we have paid special attention both to the solution of the forward problem and to the optimization algorithm. To simplify the forward problem we make use of paraxial approximations of the wave equation. Efficiency in the optimization algorithm is obtained by an exact calculation of the gradient by means of the adjoint state technique and by an adequate preconditioning. Doing so the above mentioned improvement is obtained at reasonable cost.

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.

An important research topic for prospecting seismology is to provide a fast accurate velocity model from pre-stack depth migration. Aiming at such a problem, we propose a quadratic precision generalized nonlinear global optimization migration velocity inversion. First we discard the assumption that there is a linear relationship between residual depth and residual velocity and propose a velocity model correction equation with quadratic precision which enables the velocity model from each iteration to approach the real model as quickly as possible. Second, we use a generalized nonlinear inversion to get the global optimal velocity perturbation model to all traces. This method can expedite the convergence speed and also can decrease the probability of falling into a local minimum during inversion. The synthetic data and Marmousi data examples show that our method has a higher precision and needs only a few iterations and consequently enhances the practicability and accuracy of migration velocity analysis (MVA) in complex areas.

We derive the finite‐frequency sensitivity kernel which relating the errors in migration velocity model to residual moveout in common image gathers obtained from plane‐wave source prestack depth migration. The broadband sensitivity kernel is validated by the directly measured sensitivity map. Based on this kernel, we propose a new wave‐equation migration velocity analysis approach particularly for plane‐wave source migration. This method avoids the expensive local angle or offset decomposition. Numerical method based on the one‐way propagator is designed to calculate the sensitivity kernel. Using a simple 2D synthetic data set, we demonstrate the potential application of this new method in migration velocity analysis.

Summary An efficient prestack depth migration, called ‘parsimonious’ migration was developed isotropic media a decade ago. The common-reflection point (CRP) migration velocity analysis was developed later for isotropic media. In this paper, both methods are extended for application to 2D tilted transversely isotropic (TTI) media and illustrated with synthetic data. The extension to TTI media requires redevelopment of each of the numerical components. The model parameterization consists of Thomsen’s parameters. The common-reflection point migration velocity analysis for TTI media for P-waves includes (and inverts for) Vpo, ε and δ. The orientation of the anisotropic symmetry axis have to be constrained. If it constrained orthogonal to the layer bottom, it is estimated at each CRP and updated at each iteration without intermediate picking. The extension to TTI media requires development of a new inversion procedure. The TTI CRP MVA is applied to a version of the TTI BP model by layer stripping. Vpo, δ and the orientation of the anisotropic symmetry axis (constrained orthogonal to the local reflector orientation) are successfully inverted. ε is less well constrained by the small acquisition aperture in the data. Previously published as part of Oropeza, Ernesto V., and George A. McMechan, 2014, Common-reflection-point migration velocity analysis of 2D P-wave data from TTI media. Geophysics, 79, no. 3, http://dx.doi.org/10.1190/geo2013-0295.1. Reproduced as abstract for EAGE with permission from SEG.

Prestack depth migration provides a powerful tool for velocity analysis in complex media. Both prominent approaches to velocity analysis--depth-focusing analysis and residual-curvature analysis, rely on approximate formulas to update velocity. Generally, these formulas are derived under the assumptions of horizontal reflector, lateral velocity homogeneity, or small offset. Therefore, the conventional methods for updating velocity lack accuracy and computational efficiency when velocity has large, lateral variations. Here, based on ray theory, I find the analytic representation for the derivative of imaged depths with respect to migration velocity. This derivative function characterizes a general relationship between residual moveout and residual velocity. Using the derivative function and the perturbation method. I derive a new formula to update velocity from residual moveout. In the derivation, I impose no limitation on offset, dip, or velocity distribution. Consequently, I revise the residual-curvature-analysis method for velocity estimation in the postmigrated domain. Furthermore, my formula provides sensitivity and error estimation for migration-based velocity analysis, which is helpful in quantifying the reliability of the estimated velocity. The theory and methodology in this paper have been tested on synthetic data (including the Marmousi data).