Conventional semblance methods can be avoided for a number of imaging tasks if local slopes can be directly extracted from prestack data, for example, by filtering schemes. Recent literature shows a revival of the idea for various purposes, such as velocity analysis, tau-p imaging, migration to zero offset and time migration. Here, we discuss several different ways of extracting the desired slope information from the data. We propose a simple, straightforward correction to linear planewave destructors. The correction is based on the observation that additionally to the local slope, also its inverse can be extracted from the data in a fully analogous way. Combining the information of both extractions yields a simple but powerful correction to the local slopes. In our numerical examples, the naive application of simple linear plane-wave destructors with our simple, straightforward correction produced results of high quality, even in an example with a rather high noise level and interfering events.
The estimation of kinematic attributes of locally coherent events in seismic data or seismic images, is an essential step for several recent developments in seismic data processing and velocity model building. Perhaps, the most visible ones are those connected with seismic tomography in which, not only traveltimes but also slowness components of events and possible other time-domain attributes are used for velocity model building. Famous examples are stereotomography (Billette and Lambaré, 1998; Billette et al., 2003) and NIP-wave tomography (Duveneck, 2004). Locally coherent events are also applied to velocity-independent time imaging (Fomel, 2007b).
The estimation of kinematic attributes is usually performed in two steps. The first one is a detection step based on local coherence analysis and the second one is an extraction step based on the coherence level and continuity of the event. Here, we investigate different implementations of plane-wave destructors for automatic detection of locally coherent events.
Two implementations use small moving windows through data. In the first algorithm, a single slope at the center of the window is computed by linear least squares. The second algorithm implements the prediction-error filter approach proposed by Fomel (2002). We compare these moving-window strategies with a global inversion of the slope field proposed by Fomel (2002). The global slope estimation admits different alternatives of smoothing the slowness field (Fomel, 2007a) by regularization, but is computationally demanding. The slope estimation using local windows is computationally very fast in comparison with the global estimation alternative and less dependent on prior information.
We present numerical experiments using the local and global strategies for slope estimation on a simple synthetic data, corrupted by white noise. These initial results suggest that the estimation of event slopes using local windows can be a very efficient alternative to the detection of locally coherent events.
The extraction of local slopes is done by so-called plane-wave destructors (Fomel, 2002; Claerbout, 2004). Our goal is to estimate the local slope p(x, t) for any seismic section which, in general, containing not only plane-wave events but also curved ones. The first approach is basically the technique presented in Claerbout(2004).
The quality of seismic images obtained by reverse time migration strongly depend on the employed image condition. We propose a new imaging condition, which is motivated by stationary phase analysis of the classical cross-correlation imaging condition. Its implementation requires the Poynting vector of the source and receiver wavefields at the imaging point. An obliquity correction is added to compensate for the reflector dip effect on amplitudes of reverse time migration. Numerical experiments show that using an imaging condition with obliquity compensation improves reverse time migration by reducing backscattering artifacts and improving the illumination compensation.
Pre-stack reverse time migration (RTM) is based on the time reversal property of the two-way wave equation and the crosscorrelation imaging condition proposed by Claerbout (1985). Several implementations of RTM using this imaging condition have been reported (McMechan, 1983; Kosloff and Baysal, 1983; Baysal et al., 1983).
The computational demand for RTM is high compared to wave equation migration by downward extrapolation of the wavefield (Biondi, 2006). However, low cost parallel computing and more efficient storage hardware is making RTM feasible. The difficulties of seismic imaging below the salt column and in areas of high lateral velocity variation have drawn attention to RTM, which, at least theoretically, is able to meet those challenges.
RTM has its limitations, though. Two major drawbacks are the artifacts produced by backscattering and the amplitudes of the migrated images, which are not proportional to the subsurface reflectivity (Biondi, 2006). To reduce the artifacts due to backscattering, several approaches have been recently proposed. Guitton et al. (2007) use a least-squares regularization; Fletcher et al. (2005) introduced a new forward waveequation to attenuate backscattering events and Yoon and Marfurt (2006) introduced the Poynting vector imaging condition. Several attempts to improve the amplitudes in RTM are based on illumination compensation with different kinds of regularization (Valenciano and Biondi, 2003; Kaelin and Guitton, 2006). Attempting to better understand the amplitudes in RTM, Haney et al. (2005) performed an asymptotic analysis of the crosscorrelation imaging condition. Their analysis assumes a single planar reflector in a 3D homogeneous medium, full coverage, and infinite aperture. They demonstrate that the amplitudes of RTM are affected by an obliquity factor that depends on the reflector dip. Based on this result, we propose an imaging condition which can asymptotically correct for this obliquity factor in RTM. We present numerical experiments that show the improvement of RTM images when the obliquity and illumination compensation are applied in the imaging condition.
Numerical experiments demonstrate the improvement of the images when the obliquity factor and illumination compensation are included in the imaging condition for RTM. Here, we use a finite-difference implementation, but the correction can also be incorporated into Fourier-domain implementations like the one of Tessmer (2003).
We start by revisiting the asymptotic analysis of the crosscorrelation imaging condition (Haney et al., 2005). Based on this result we propose an imaging condition for RTM which, asymptotically, improves the amplitude of RTM images. Asymptotic analysis of cross-correlation imaging condition
The cross-correlation imaging condition for shot-profile migration (Claerbout, 1985)
Seismic imaging in depth is limited by the accuracy of velocity model estimation. Slope tomography uses the slowness components and traveltimes of picked reflection or diffraction events for velocity model building. The unavoidable data incompleteness requires additional information to assure stability to inversion. One natural constraint for ray based tomography is a smooth velocity model. We propose a new, reflectionangle- based kind of smoothness constraint as regularization in slope tomography and compare its effects to three other, more conventional constraints. The effect of these constraints are evaluated through comparison of the inverted velocity models as well as the corresponding migrated images. We find the smoothness constraints to have a distinct effect on the velocity model but a weaker effect on the migrated data. In numerical tests on synthetic data, the new constraint leads to geologically more consistent models.
Slope tomography is one of the many methods that try to determine a macrovelocity model for time or depth imaging. It uses slowness vector components to improve and stabilize the traveltime inversion. Slope tomography was initially proposed by Billette and Lambaré (1998) as a robust tomographic method for estimating velocity macro models from seismic reflection data. They had recognized the potential efficiency of traveltime tomography (Bishop et al., 1985; Farra and Madariaga, 1988) but also the difficulties associated with a highly interpretative picking. The selected events have to be tracked over a large extent of the pre-stack data cube, which is quite difficult for noisy or complex data. The idea is to use locally coherent events characterized by their slopes in the pre-stack data volume. Such events can be interpreted as pairs of ray segments and provide independent information about the velocity model.
However, the data for slope tomography are incomplete (Bishop et al., 1985). This causes depth and velocity ambiguities that depend strongly on the size of the acquisition aperture (Bube et al., 2005). Therefore, stability and convergence can only be achieved if additional information is prescribed. This additional information contains desirable properties for the solution, reducing ambiguity (Menke, 1989). It can be shown that stability is obtained only if we try to determine a smooth model of the subsurface (Delprat-Jannaud and Lailly, 1992, 1993). Moreover, for ray based inversion, smoothness is a requirement, because rough models cause the forward problem to break down during linear iterations. The use of combined smoothness constraints enables an interpretation-oriented inversion while keeping solutions consistent with the data.
We investigate the effect of different kinds of smoothness constraints in slope tomography, prescribing lateral, vertical and isotropic smoothing constraints in different combinations. Moreover, we propose a structurally motivated smoothing constraint in the direction of a potential reflector. This regularization is based on information that is contained in the data, in contrast to standard regularizations that impose global smoothness constraints. We test the different regularizations on the Marmousoft data set (Billette et al., 2003).
Slope tomography differs from conventional reflection tomography by the data that are used for the inversion (Billette et al., 2003).
Image-wave propagation in the common-image gather (CIG) domain can be combined with residual-moveout analysis for iterative migration velocity analysis. This leads to a detection of those velocities where events flatten. The procedure can be applied in inhomogeneous media. For this purpose, the CIGs obtained by migration with an inhomogeneous macrovelocity model are continued starting from a constant reference velocity. The interpretation of continued CIGs as obtained from residual migrations leads to a correction formula that translates the residual flattening velocities into absolute time-migration velocities. In this way, the migration velocity model can be iteratively improved until a satisfactory result is reached. By means of a numerical example, we show that migration velocity analysis with iterative image continuation is able to construct a migration velocity model from scratch.
The quality of a final seismic depth or time migrated image depends on the available velocity information. Generally, a macrovelocity model obtained from conventional velocity analysis needs to be improved with migration velocity analysis (MVA).
In MVA, different migrated images of the same position obtained from different subsets of the data are collected into socalled common-image gathers (CIGs). The CIGs are sensitive to the velocity model (Al-Yahya, 1989). The idea is that for the correct migration velocity, the image of the same subsurface point should be independent of the data subset used to produce it. Thus, the reflection events in a CIG should be flat regardless of the structures (Stork, 1992). On the other hand, for a incorrect migration velocity, different data subsets will lead to different images, thus resulting in nonflat reflection events in the CIG (see also Zhu et al., 1998). Because of its conceptual clearness, residual-moveout (RMO) analysis in the offset domain (Al-Yahya, 1989) or angle domain (Biondi and Symes, 2004) has become the favorite tool for MVA. During the years, many improvements have been suggested (see, e.g., references in Fei and McMechan, 2006).
Another approach to image updating is based on the continuation of a seismic image in velocity. This is a straightforward extension of residual migration (Rothman et al., 1985) or cascaded migration (Larner and Beasley, 1987) to continuous velocity variation (Claerbout, 1986; Fomel, 1994). The kinematics and dynamics of image continuation in the time-migrated domain have been thoroughly discussed by Fomel (2003b). In a companion paper, Fomel (2003a) suggested its use for timemigration velocity analysis in the prestack domain.
When a seismic section is depth- or time-migrated with different (constant) migration velocities, different reflector images of the subsurface are obtained. By continuously changing the migration velocity, the image can be continued in velocity. Based on the kinematics of the dislocation of the image of a seismic event under variation of the migration velocity, Fomel (1994) presented partial differential equations that describe this velocity continuation. Due to the similarity with propagating waves, Hubral et al. (1996) interpreted the continued images for different velocities as snapshots of propagating "image waves". Consequently, they termed the resulting differential equations "image-wave equations".