A new imaging technique for reliable migration velocity analysis

In case Of complex StmCtures involving important lateral velocity variations individual prestack depth migrated images are not interpretable: yet, migration coherency panels do not allow to check the con~%~~ of the velocity model. We propose to substitute prestack depth migration by Prestack imaging by coupled linearized inversion (PICL.1). First results obtained with this new imaging method show that it indeed nsolves the difficulty.

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.

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.

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.

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

The determination of small‐scale velocity anomalies (from tens to a few hundreds of metres) is a major problem in seismic exploration. The impact of such anomalies on a structural interpretation can be dramatic and conventional techniques such as tomographic inversion or migration velocity analysis are powerless to resolve the ambiguity between structural and velocity origins of anomalies. We propose an alternative approach based on stochastic modelling of numerous anomalies until a set of models is found which can explain the real data. This technique attempts to include as much a priori geological information as possible. It aims at providing the interpreter with a set of velocity anomalies which could possibly be responsible for the structural response. The interpreter can then choose one or several preferred models and pursue a more sophisticated analysis. The class of retained models are all equivalent in terms of data and therefore represent the uncertainty in the model space.The procedure emulates the real processing sequence using a simplified scheme. Essentially, the technique consists of five steps: 1 Interpretation of a structural anomaly in terms of a velocity anomaly with its possible variations in terms of position, size and amplitude. 2 Drawing a model by choosing the parameters of the anomaly within the acceptable range. 3 Modelling the traveltimes in this model and producing the imaging of the reflected interface. 4 Comparing the synthetic data with the real data and keeping the model if it lies within the data uncertainty range. 5 Iterate from step 2.In order to avoid the high computational cost inherent in using statistical determinations, simplistic assumptions have been made:• The anomaly is embedded in a homogeneous medium: we assume that the refraction and the time shift due to the anomaly have a first‐order effect compared with ray bending in the intermediate layers.• We model only the zero‐offset rays and therefore we restrict ourselves to structural problems.• We simulate time migration and so address only models of limited structural complexity.These approximations are justified in a synthetic model which includes strong lateral velocity variations, by comparing the result of a full processing sequence (prestack modelling, stack and depth migration) with the simplified processing. This model is then used in a blind test on the inversion scheme.

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.

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.

Through the use of iterative turning‐ray tomography followed by wave‐equation datuming (or tomo‐datuming) and prestack depth migration, we generate accurate prestack images of seismic data in overthrust areas containing both highly variable near‐surface velocities and rough topography. In tomo‐datuming, we downward continue shot records from the topography to a horizontal datum using velocities estimated from tomography. Turning‐ray tomography often provides a more accurate near‐surface velocity model than that from refraction statics. The main advantage of tomo‐datuming over tomo‐statics (tomography plus static corrections) or refraction statics is that instead of applying a vertical time‐shift to the data, tomo‐datuming propagates the recorded wavefield to the new datum. We find that tomo‐datuming better reconstructs diffractions and reflections, subsequently providing better images after migration. In the datuming process, we use a recursive finite‐difference (FD) scheme to extrapolate wavefield without applying the imaging condition, such that lateral velocity variations can be handled properly and approximations in traveltime calculations associated with the raypath distortions near the surface for migration are avoided. We follow the downward continuation step with a conventional Kirchhoff prestack depth migration. This results in better images than those migrated from the topography using the conventional Kirchhoff method with traveltime calculation in the complicated near surface. Since FD datuming is only applied to the shallow part of the section, its cost is much less than the whole volume FD migration. This is attractive because (1) prestack depth migration usually is used iteratively to build a velocity model, so both efficiency and accuracy are important factors to be considered; and (2) tomo‐datuming can improve the signal‐to‐noise (S/N) ratio of prestack gathers, leading to more accurate migration velocity analysis and better images after depth migration. Case studies with synthetic and field data examples show that tomo‐datuming is especially helpful when strong lateral velocity variations are present below the topography.

This article, written by Special Publications Editor Adam Wilson, contains highlights of paper IPTC 18293, “Near-Surface Velocity Model of Dukhan Field From Multiphysics Survey To Enhance PSDM Seismic Imaging,” by K. Setiyono, Qatar Petroleum; S. Gallo, C. Boulanger, F. Bruere, F. Moreau, and B. Rondeleux, CGG; and J. Snow, Qatar Petroleum, prepared for the 2015 International Petroleum Technology Conference, Doha, Qatar, 7-9 December. The paper has not been peer reviewed. Dukhan field affords an opportunity to assess alternative near-surface velocity-modeling work flows that specifically integrate microgravity (MGR) and vertical electrical sounding (VES). A pilot study of the Dukhan seismic data demonstrates that integration of MGR/VES into prestack depth migration (PSDM) improves the fidelity of seismic images and results in an improved depth match. The resultant MGR/VES PSDM pilot model improved the velocity model between the surface and top Simsima formation on the basis of visible impact on the deeper structure obtained from imaging and better fit between the seismic depth horizons and well measured depths. Introduction The Dukhan field (Fig. 1) along the southwest coast of Qatar is an elongated anticline that has significant hydrocarbon reservoirs in the Jurassic Arab and deeper formations. In 2007, Qatar Petroleum invested in a new full-field, land and shallow-water 3D-seismic survey of the area. Despite considerable and continuous progress in seismic technology, obtaining an accurate, quantitative, well-focused seismic depth image at reservoir level remains challenging. In land seismic, a poor characterization of the near-surface geology is detrimental to the depth imaging of underlying structures. The proper knowledge of the near surface to correct for its perturbations onto the seismic wavefield remains one of the main challenges to be addressed. The near surface influences the kinematics as well as the dynamics of the source-generated seismic waves. Knowledge of the near surface to correct for its induced distortions is critical for a reliable use of seismic images, structures, and amplitude at reservoir level. The near-surface lateral and vertical velocity variations often are not accessible with seismic methods, even to the seismic refraction survey, because their depth of investigation is limited to the top of the shallowest hard layer. In addition, rapid lateral changes and large velocity contrasts are unfavorable to seismic refraction.

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.

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).

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.

I used the common-reflection-surface (CRS) stack technique together with conventional common-midpoint processing and prestack depth migration (PSDM) to process three parallel high-resolution seismic reflection lines acquired in a closed industrial plant. This brownfield site, now undergoing remediation, is located within the Bagnoli district of the city of Naples, Italy, near the southeastern border of Campi Flegrei caldera. Compared with the other results, the CRS stack, followed by poststack depth migration, produced a seismic image more consistent with the expected subsurface structures and with outcropping units present in the area. The survey confirmed the presence under the brownfield site of some buried steps of Coroglio Fault, a main structure formed approximately [Formula: see text] ago, during the Neapolitan Yellow Tuff supereruption. Results have important implications for hazard assessment and geothermal exploration because the imaged faults certainly are preferential passageways for geothermal fluids and washed pollutants. These tests revealed that the common-reflection-surface stack can be a fast and cost-effective alternative to PSDM in settings in which structural complexity and high levels of ambient noise make it challenging to obtain a reliable background velocity model; therefore, allowing high-resolution reflection seismology to be successfully used in those environments.

In the last 25 years, mathematical and numerical modeling of cardiovascular problems has experienced a spectacular progress. Moving progressively from the stage of idealized models supporting traditional in vitro and in vivo approaches – basically with proofs of concept – Cardiovascular Mathematics has been transforming into a unique tool for patient-specific analysis. This has been made possible by the strong integration with imaging devices. Nowadays, the work pipeline starting from a medical image in the Digital Imaging and Communications in Medicine (DICOM) format to retrieve the patient-specific geometry of an artery or the left ventricle to be used as computational domain in a numerical simulation is well established. This has initiated a paradigm shift in the interactions between mathematicians, bioengineers, and medical doctors. The latter are progressively getting used to in silico analysis as an important component of their research [3]. The introduction of numerical procedures as a part of an established clinical routine and more in general of a consolidated support to the decision-making process of physicians still requires some steps both in terms of infrastructures (to bring computational tools to the operating room or the bedside) and methods. In particular, the quality of the numerical results needs to be assessed and certified. The reliability of simulations calls for an accurate quantification and possibly reduction of uncertainty. In this scenario and in view of terrific advancements of measuring techniques, an important research line – quite established in other fields – is data assimilation [4,6,9], that is, the integration of numerical simulations and measurements. We may say that numerical models provide a background knowledge (based on physical principles and constitutive laws, not patient-specific), whereas measures give a foreground (individual) information; accuracy of in silico procedures relies on the correct integration of these two levels of knowledge of the problem [9,10]. As a matter of fact, numerical models depend on parameters and boundary/initial conditions. Their knowledge is in practice usually incomplete, inaccurate, and noisy. In some cases, the sensitivity itself of the final results to these parameters is unclear and needs to be quantified. Such an uncertainty may be significantly reduced by the availability of patient-specific measures; on the other hand, the quality of measures can be strongly enhanced by comparison with mathematical models (after all, noise is in general not educated, it does not know PDEs . . . ) [4,5,7,8,9,11]. We think therefore that, in the future, a crucial step for the clinical application of Cardiovascular Mathematics will be the passage from numerical simulation to data assimilation. To be concrete, borrowing a motto by Michel Fortin supporting mesh adaptivity in finite element simulations, stating that a ‘grid should not be considered a data, but an unknown of the problem’, we think that many parameters currently tuned on the basis of nonpatient specific criteria (even in patient-specific geometries) should be considered part of the unknowns to be found by assimilation procedures.