Moveout-based wave-equation migration velocity analysis

Prestack depth migration (PSDM) heavily depends on the accuracy of the migration velocity model. Reflection tomography on the base of common-image gathers (CIGs) is an important tool to build such models in depth domain. Unlike the conventional offset domain PSDM, angle domain PSDM can avoid migration artifacts when the velocity models are complex. It is easier to build the back-projection matrix for tomography according to the residual moveout (RMO) on the angle domain common image gathers (ADCIGs). In this abstract, we propose a novel nonlinear post-migration tomography approach to build velocity model using ADCIGs. The picked angle domain RMO is converted to prestack domain as invariants for data-fitting and model updating in the tomography iteration. Synthetic example proves the validity of this approach.

In this paper we present the improved algorithm for angle domain common imaging gathers (ADCIG) during Kirchhoff pre‐stack time migration (PSTM). The algorithm can calculate relatively true incident angle under three‐dimensional circumstance, while the traditional ray‐parameter method could not handle the incident angle for the dipping events. Compared with straight‐ray Kirchhhoff angle domain gathers method presented in recent years, our method is able to extract wide‐angle information. Increased angular illumination is beneficial for AVA inversion. The results of synthetic model data have verified that: for V(z) (layered) media, the traveltime calculation taking ray‐bending effect into account, is much more accurate than straight ray for large angle. Thus the wide angles of ADCIG are flatter than the straight assumption. Moreover, the range of ADCIG is enlarged. The ADCIG and its AVA inversion of 3‐D field data have also demonstrated the improvement at wide angle.

Angle domain common image gathers (ADCIGs) from reverse-time migration (RTM) provide new tools for imaging complex geologic structures, such as salt flank or subsalt areas, characterized by multiarrivals. Compared with common image gathers from Kirchoff or Beam migration, the ADCIGs from RTM have the advantage of relying on wave propagation, providing a more reliable input for tomography or amplitude analysis. In practice however, most current wide azimuth surveys (e.g., deep-water regions of the Gulf of Mexico) are acquired with coarsely sampled shot and receiver locations on the surface, which leads to severe angular undersampling. This phenomenon is frequently observed on shallow seismic events in high-resolution 3D ADCIGs from common-shot RTM. In addition, because small offsets are frequently not recorded, seismic events are missing at small incidence angles in angle gathers. We have made a detailed study of the angular sampling issue in 3D angle gathers. We then used plane-wave RTM to generate 3D angle gathers. Plane-wave RTM, with its low-cost and automatic angular interpolation, is a promising solution for improving the quality of 3D angle gathers.

Summary The angle-domain common image gathers (ADCIGs) describe the reflectivity variation over the reflection angles which are important for seismic exploration. However, because of the limitation of seismic frequency band and acquisition geometry, conventional reverse time migration (RTM) produces the ADCIGs with low fidelity. We propose a reflection angle-domain extended least-squares RTM (AELSRTM) to improve the quality of the ADCIGs in which the Poynting vector is used to efficiently calculate the angles. We first derive a feasible pseudo-extended linearized modeling operator which could map the ADCIGs into the seismic data. The modeling operator, associated with an adjoint operator, are then used to build an extended linearized inversion framework for inverting the angle-dependent reflectivity. Taking advantage of the simplicity and coherency of the ADCIGs, we impose a series of low-rank constraints on the extended angle dimension and a sparse constraint on the whole dimension to ensure the production of high-quality ADCIGs. We conduct numerical examples on the Marmousi model, and the results demonstrate that AELSRTM could produce the ADCIGs and the stacked image with higher fidelity than RTM, which provides a reliable input for anisotropic model building and amplitude versus angle analysis.

Common-image gathers (CIGs) are extensively used in migration velocity analysis. Any defocused events in the subsurface offset domain or equivalently nonflat events in angle-domain CIGs are accounted for revising the migration velocities. However, CIGs from wave-equation methods such as reverse time migration are often expensive to compute, especially in 3D. Using the excitation amplitude imaging condition that simplifies the forward-propagated source wavefield, we have managed to extract extended images for space and time lags in conjunction with prestack reverse time migration. The extended images tend to be cleaner, and the memory cost/disk storage is extensively reduced because we do not need to store the source wavefield. In addition, by avoiding the crosscorrelation calculation, we reduce the computational cost. These features are demonstrated on a linear [Formula: see text] model, a two-layer velocity model, and the Marmousi model.

We present two methods to compute angle-domain common image gathers (ADCIG) by downward-continuation migration, and we an-alyze their amplitude response versus reflection angle (AVA). A straightforward implementation of the two methods leads to contra-dictory, and thus obviously inaccurate, amplitude responses. The amplitude problem is related to the fact that downward continua-tion migration is the adjoint of upward-continuation modeling, but it is a poor approximation of its inverse. We derive the weight-ing operators, diagonal in the frequency-wavenumber domain, that makes migration a good approximation to the inverse of modeling. After weighting, the ADCIGs computed by the two methods be-come consistent. Other important factors degrading the accuracy of AVA in practical situation are the limited sampling and offset range, and the bandlimited nature of seismic data.

True-amplitude wave-equation migration provides a quality migrated image of the earth's interior. In addition, the amplitude of the output provides an estimate of the angular-dependent reflection coefficient, similar to the output of Kirchhoff inversion. Recently, true-amplitude wave-equation migration for common-shot data has been proposed to generate amplitude-reliable, shot-domain, common-image gathers in heterogeneous media. We present a method to directly produce angle-domain common-image gathers from both common-shot and shot-receiver wave-equation migration. Generating true-amplitude, shot-domain, common-image gathers requires a deconvolution-type imaging condition using the ratio of the upgoing and downgoing wavefield, each downward-projected to the image point. Producing true-amplitude, angle-domain, common-image gathers requires, instead, the product of the upgoing wavefield and the complexconjugate of the downgoing wavefield in the imaging condition. Since multiplication is a more stable computational process than division, the new methods proposed provide more stable ways of inverting seismic data. Furthermore, the resulting common-image gathers can be directly used for migrated amplitude-variation-with angle analysis and tomography-based velocity analysis. Shot-receiver wave-equation migration requires new true-amplitude, one-way wave equations with one depth variable and transverse variables for the coordinates corresponding to sources and receivers, hence, two transverse coordinates in 2D and four transverse coordinates in 3D. We propose a modified double-square-root one-way wave equation to produce true amplitude common-image angle gathers. We also demonstrate the new methods with some synthetic examples. Some numerical examples show that the new methods we propose give better amplitude performance on the migrated angle gathers.

Wave-equation migration velocity analysis (WEMVA) is a non-linear optimization process that aims at estimating seismic velocities from reflection data using the wave equation. One important issue in Born-approximation-based WEMVA is preparation of an image perturbation free of cycle-skipping during each linear update. I show that this step can be accomplished by estimating the non-stationary slope of defocused events in subsurface offset gathers (ODCIG). The analysis also compares and unifies two popular WEMVA approaches, namely differential semblance optimization (DSO) and differential residual migration (DRM). The proposed method could be considered as an improved DSO implementation that overcomes some of its limitations by incorporating concepts from DRM.

Wavelength-dependent Fresnel beam propagator and migration in VTI media

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

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.

The ideal approach for continuous reservoir monitoring allows generation of fast and accurate images to cope with the massive data sets acquired for such a task. Conventionally, rigorous depth-oriented velocity-estimation methods are performed to produce sufficiently accurate velocity models. Unlike the traditional way, the target-oriented imaging technology based on the common-focus point (CFP) theory can be an alternative for continuous reservoir monitoring. The solution is based on a robust data-driven iterative operator updating strategy without deriving a detailed velocity model. The same focusing operator is applied on successive 3D seismic data sets for the first time to generate efficient and accurate 4D target-oriented seismic stacked images from time-lapse field seismic data sets acquired in a [Formula: see text] injection project in Saudi Arabia. Using the focusing operator, target-oriented prestack angle domain common-image gathers (ADCIGs) could be derived to perform amplitude-versus-angle analysis. To preserve the amplitude information in the ADCIGs, an amplitude-balancing factor is applied by embedding a synthetic data set using the real acquisition geometry to remove the geometry imprint artifact. Applying the CFP-based target-oriented imaging to time-lapse data sets revealed changes at the reservoir level in the poststack and prestack time-lapse signals, which is consistent with the [Formula: see text] injection history and rock physics.

A methodology for velocity estimation using reverse time migration (RTM) is described. An objective function is posed in the image domain that promotes image focusing and minimizes non-focused energy. We use the adjoint state method for the gradient calculation. The procedure of gradient calculation is similar to RTM. The inversion uses amplitude normalization to mitigate the effects of the amplitude fluctuation in image gathers. Synthetic data examples demonstrate that the method can recover long wave-length information as well as some of the finer-scale details in the velocity model. It is robust in that the convergence does not require a starting model that contains the low frequency velocity structure.

Angle domain common image gathers (ADCIGs) are the basic data for the MVA and AVA. In this paper, a common angle gather output scheme of Kirchhoff PSDM is introduced. It includes: (1) the traveltime calculation is based on the dynamic programming approach adapted to arbitrary medium, (2) obtaining the reflection angle i according to the gradient of traveltime filed in the image space, (3) ADCIGs are extracted in the migration process. We compare the offset domain common image gathers(ODCIGs) of numerical models with the ADCIGs of numerical models. Numerical examples on synthetic data demonstrate the feasibility of the method.

Tomograhic model building has commonly been conducted by using offset gathers from the prestack migration. However, in complex media, common offset images may yield an inconsistent image of the subsurface and thus cause problems in the model building process. The angle domain common image gathers (ADCIGs), on the other hand, do not suffer from such inconsistency and open a new opportunity for model building. The same model updating steps in the offset domain can be followed in angle domain with a different picking tool and a slightly different equation set up procedure. The residual picking is done robustly and efficiently using a single parameter curvature analysis. The angle gather ray tracing is more efficient than offset gather ray tracing. The method can be applied to angle gathers from ray-based or wave equation imaging. The field data test shows that this new work flow can provide a model that improves the base of salt and images below the salt.