Presence Of Lateral Velocity Variations Research Articles

Surface wave analysis for heterogeneous geological formations in geothermal fields: effect of wave propagation direction

ABSTRACTWe applied advanced surface wave analysis for multichannel and multishot seismic data to estimate S-wave velocity structure with high lateral resolution. Although a horizontally layered structure is typically assumed in surface wave analysis, this assumption might be violated in environments such as geothermal fields because of their heterogeneous geological formation. The lateral variation of phase velocity can be effectively estimated with common midpoint (CMP) cross-correlation (CMPCC) analysis. In this study, we introduced two additional approaches into the CMPCC analysis workflow, to further improve lateral resolution of phase velocity estimates. One approach is window-controlled CMPCC analysis, which applies a spatial window for the CMP gathers while maintaining the accuracy of the phase velocity estimates. In this analysis, we found that it is difficult to improve phase velocity estimates at lower frequencies, as a wider spatial window must be kept to maintain the accuracy of the dispersion curve. Therefore, we introduced another approach in which we consider the direction of surface wave propagation from sources to receivers. Selection of cross-correlations based on source–CMP direction provides a significant improvement in dispersion curve resolution in the presence of lateral velocity variations, for a wide frequency range. We applied direction-controlled CMPCC analysis to seismic data acquired in the Yamagawa geothermal field, Kyushu Island, southwest Japan, and obtained dispersion curves in the heterogeneous geological setting. We then obtained S-wave velocity profiles by applying genetic algorithm inversion to the dispersion curves. The S-wave velocity profiles in the geothermal field resolve shallow and local heterogeneous structures (e.g. volcanic ash, pumice and igneous intrusions) that cannot be identified on reflection seismic profiles.

A robust approach to time‐to‐depth conversion and interval velocity estimation from time migration in the presence of lateral velocity variations

ABSTRACTThe problem of conversion from time‐migration velocity to an interval velocity in depth in the presence of lateral velocity variations can be reduced to solving a system of partial differential equations. In this paper, we formulate the problem as a non‐linear least‐squares optimization for seismic interval velocity and seek its solution iteratively. The input for the inversion is the Dix velocity, which also serves as an initial guess. The inversion gradually updates the interval velocity in order to account for lateral velocity variations that are neglected in the Dix inversion. The algorithm has a moderate cost thanks to regularization that speeds up convergence while ensuring a smooth output. The proposed method should be numerically robust compared to the previous approaches, which amount to extrapolation in depth monotonically. For a successful time‐to‐depth conversion, image‐ray caustics should be either nonexistent or excluded from the computational domain. The resulting velocity can be used in subsequent depth‐imaging model building. Both synthetic and field data examples demonstrate the applicability of the proposed approach.

Moveout analysis of wide-azimuth data in the presence of lateral velocity variation

Moveout analysis of wide-azimuth reflection data seldom takes into account lateral velocity variations on the scale of spreadlength. However, velocity lenses (such as channels and reefs) in the overburden can cause significant, laterally varying errors in the moveout parameters and distortions in data interpretation. Here, we present an analytic expression for the normal-moveout (NMO) ellipse in stratified media with lateral velocity variation. The contribution of lateral heterogeneity (LH) is controlled by the second derivatives of the interval vertical traveltime with respect to the horizontal coordinates, along with the depth and thickness of the LH layer. This equation provides a quick estimate of the influence of velocity lenses and can be used to substantially mitigate the lens-induced distortions in the effective and interval NMO ellipses. To account for velocity lenses in nonhyperbolic moveout inversion of wide-azimuth data, we propose a prestack correction algorithm that involves computation of the lens-induced traveltime distortion for each recorded trace. The overburden is assumed to be composed of horizontal layers (one of which contains the lens), but the target interval can be laterally heterogeneous with dipping or curved interfaces. Synthetic tests for horizontally layered models confirm that our algorithm accurately removes lens-related azimuthally varying traveltime shifts and errors in the moveout parameters. The developed methods should increase the robustness of seismic processing of wide-azimuth surveys, especially those acquired for fracture-characterization purposes.

A Fourier integral algorithm and its GPU/CPU collaborative implementation for one-way wave equation migration

Pre-stack one-way wave equation (OWE) is a useful tool for seismic imaging and modeling. Since the idea of OWE appeared in the 1970s, geophysicists have made great effort to improve the accuracy of the one-way wave equation extrapolators. In this paper, we present the idea of solving OWE using the Fourier integral method, which represents OWE as a Fourier integral equation and solves it in dual spaces (both space and wave-number domains). By doing this, we can propagate wave-fields up to nearly 90° angle from the vertical direction in the presence of lateral velocity variations. The proposed method is stable, does not suffer from the numerical dispersion, and overcomes the azimuthal anisotropy problem when extended to three dimensions. The computation cost of the Fourier integral method is too high and was considered impractical for a conventional computer. In this paper, we take advantages of the Graphic Processing Unit (GPU) and use the matrix multiplication technique to accelerate the algorithm. The speedup ratio we obtained is tens of hundreds times so that the method can be applied to a real project for pre-stack depth imaging.

Wavefield dependency on virtual shifts in the source location

ABSTRACTThe wavefield dependence on a virtual shift in the source location can provide information helpful in velocity estimation and interpolation. However, the second‐order partial differential equation (PDE) that relates changes in the wavefield form (or shape) to lateral perturbations in the source location depends explicitly on lateral derivatives of the velocity field. For velocity models that include lateral velocity discontinuities this is problematic as such derivatives in their classical definition do not exist. As a result, I derive perturbation partial differential wave equations that are independent of direct velocity derivatives and thus, provide possibilities for wavefield shape extrapolation in complex media. These PDEs have the same structure as the wave equation with a source function that depends on the background (original source) wavefield. The solutions of the perturbation equations provide the coefficients of a Taylor's series type expansion for the wavefield. The new formulas introduce changes to the background wavefield only in the presence of lateral velocity variation or in general terms velocity variations in the perturbation direction. The accuracy of the representation, as demonstrated on the Marmousi model, is generally good.

Acoustic wavefield evolution as a function of source location perturbation

SUMMARY The wavefield is typically simulated for seismic exploration applications through solving the wave equation for a specific seismic source location. The direct relation between the form (or shape) of the wavefield and the source location can provide insights useful for velocity estimation and interpolation. As a result, I derive partial differential equations that relate changes in the wavefield shape to perturbations in the source location, especially along the Earth’s surface. These partial differential equations have the same structure as the wave equation with a source function that depends on the background (original source) wavefield. The similarity in form implies that we can use familiar numerical methods to solve the perturbation equations, including finite difference and downward continuation. In fact, we can use the same Green’s function to solve the wave equation and its source perturbations by simply incorporating source functions derived from the background field. The solutions of the perturbation equations represent the coefficients of a Taylor’s series type expansion of the wavefield as a function of source location. As a result, we can speed up the wavefield calculation as we approximate the wavefield shape for sources in the vicinity of the original source. The new formula introduces changes to the background wavefield only in the presence of lateral velocity variation or in general terms velocity variations in the perturbation direction. The approach is demonstrated on the smoothed Marmousi model.

A self-adaptive algorithm for choosing reference velocities in the presence of lateral velocity variations

Based on perturbation theory, the wave equation extrapolation operator with mixed domains has the ability to deal with lateral velocity variations. It is the image method that has undergone much research in seismology. All extrapolation operators face the problem of choosing the reference velocity due to continuation in depth. The wavefield extrapolation operator with a single reference velocity is suitable for media with weak lateral variation. The multi-reference velocity extrapolation operator can cope with severe lateral velocity variations and improve image accuracy. However, the calculation cost is large. We present a self-adaptive approach to automatically determine the number of selected reference velocities according to the complexity of structure and the given velocity threshold value. The approach can be used to construct the SSF, FFD, WXFD, and GSP multi-reference velocity wavefield extrapolation image algorithms. The result of a salt-dome model data test demonstrates that the self-adoptive multi-reference wavefield extrapolation algorithm has the ability to deal with severe lateral velocity variations and can also be used for structure edge detection. The method is flexible and computationally cost-effective.

Tomographic inversion of common image gathers

Interpretation of seismic reflection data in complex geological areas requires good seismic imaging which can only be provided if the velocity field is known with high accuracy. In the presence of lateral velocity variations, pre-stack depth migration (PSDM) is indispensable for obtaining the real depths and geometries of the structures. Stacking velocities are inadequate to perform this step of the data processing, because they are evaluated supposing flat horizons and lateral constant velocity in each common depth point (Yilmaz, 2001). To avoid this problem, it is necessary to use a more accurate velocity analysis, such as the tomographic inversion technique. The tomographic velocity field is usually used to obtain the travel time table to perform PSDM. A satisfactory migration result can provide information about the reliability of the velocity field. In fact, if the velocity field is correct, the reflected events in the common image gathers (CIGs), in the pre-stack depth domain, are flats. In particular, if we use a velocity field higher than the real velocity, the reflected events become deeper versus offset while, if the velocity is lower, the reflected events become shallower versus offset. In the last decade, many efforts have been developed to determine a more detailed velocity field by using the output of PSDM. In fact, as pointed out by Lafond and Levander (1993), pre-stack migration produces a wealth of data from which it is possible to extract velocity corrections. These can then in turn be used to improve the focusing of the reflectors. The method is based on a pre-stack layer-stripping Kirchhoff algorithm with geometrical ray tracing in heterogeneous media. In this way, they iteratively update the velocity model and depth focusing of the interfaces. The analysis of the semblance of the CIG is used to upgrade the depth velocity model, if the reflected events are not flat - see, for example, Liu (1995). This procedure has the same limitations as the stacking velocity analysis, because it assumes that the velocity field is laterally constant inside the CIG. Kosloff et al. (1996) determined the velocity and the depth interface by using the tomography of migrated common reflecting point (CRP) gathers. This method is derived from the tomographic principle that relates travel time change along a given ray to perturbations in slowness and layer depths. The tomographic principle is used to convert depth errors in migrated CRP gathers to time errors along the CRP ray pair, and thus the conventional travel time tomography was used. This last method has the advantage that the migrated gathers are used at once as opposed to being analyzed at single stations. On the other hand, many PSDMs have to be performed to obtain the final model. For this reason, we developed a procedure to update the velocity field using a tomographic inversion of the picked reflected events in the CIGs which requires only one PSDM for each layer. The aim is to obtain a flatness of the picked events in every CIG at the end of our procedure. In this paper, we present the methodology, and a few applications with both synthetic and real data.

Seismic depth migration with the Dirac equation

We demonstrate the applicability of the Dirac equation in seismic wavefield extrapolation by presenting a new explicit one‐way prestack depth migration scheme. The method is in principle accurate up to 90° from the vertical, and it tolerates lateral velocity variations. This is achieved by performing the extrapolation step of migration with the Dirac equation, implemented in the space‐frequency domain. The Dirac equation is an exact linearization of the square‐root wave equation and is equivalent to keeping infinitely many terms in a Taylor series or continued‐fraction expansion of the square‐root operator. An important property of the new method is that the local velocity and the spatial derivatives decouple in separate terms within the extrapolation operator. Therefore, we do not need to precompute and store large tables of convolutional extrapolator coefficients depending on velocity. The main drawback of the explicit scheme is that evanescent energy must be removed at each depth step to obtain numerical stability. We have tested two numerical implementations of the migration scheme. In the first implementation, we perform depth stepping using the Taylor series approximation and compute spatial derivatives with high‐order finite difference operators. In the second implementation, we perform depth stepping with the Rapid expansion method and numerical differentiation with the pseudospectral method. The imaging condition is a generalization of Claerbout’s U / D principle. For both implementations, the impulse response is accurate up to 80° from the vertical. Using synthetic data from a simple fault model, we test the depth migration scheme in the presence of lateral velocity variations. The results show that the proposed migration scheme images dipping reflectors and the fault plane in the correct positions.

Plumes: Response of time migration to lateral velocity variation

We analyze how time migration mispositions events in the presence of lateral velocity variation by examining the impulse response of depth modeling followed by time migration. By examining this impulse response, we lay the groundwork for the development of a remedial migration operator that links time and depth migration. A simple theory by Black and Brzostowski predicted that the response of zero‐offset time migration to a point diffractor in a v(x, z) medium would be a distinctive, cusp‐shaped curve called a plume. We have constructed these plumes by migrating synthetic data using several time‐migration methods. We have also computed the shape of the plumes by two geometrical construction methods. These two geometrical methods compare well and explain the observed migration results. The plume response is strongly influenced by migration velocity. We have studied this dependency by migrating synthetic data with different velocities. The observed velocity dependence is confirmed by geometrical construction. A simple first‐order theory qualitatively explains the behavior of zero‐offset time migration, but a more complete understanding of migration velocity dependence in a v(x, z) medium requires a higher order finite‐offset theory.

Laterally Varying Velocity Estimation

Lateral velocity variations in the subsurface are a departure from the simple horizontally layered model on which the extraction of NMO velocities and interval velocities are based. Significant differences between stacking velocity and interval velocity may result in complex geological areas when the estimation is based on this horizontally layered media assumption. In this paper methods are presented of velocity estimation that are both quick and accurate in the presence of lateral velocity variations. The method developed here solves an analytic expression by fitting an unknown smooth velocity function, and also determines the reflector depth using only two simple assumptions: (1) that the ray in the laterally varying medium travels along the same path as in the homogeneous medium; and (2) that the subsurface velocity can be approximated by a smoothly varying lateral velocity function. Assumption (1) is shown to introduce only higher order correction terms to the travel time for a given source-receiver pair. It is demonstrated by assumption (2) how the use of a stable, smooth and continuous velocity function across the section results in a better migration velocity than a velocity analysis performed at an isolated CDP location. By estimating the degree of lateral velocity variation across the seismic section at an early stage in the processing sequence, a decision on whether to use pre- or post-stack migration can be made with reliable velocity information. This method has the advantage of using an analytic expression for the travel time instead of complicated ray tracing by solving differential equations. This makes it useable in an interactive velocity analysis mode. An example using field data from the North West Shelf of Australia shows how the new travel time equation can improve the velocity analysis over a fault zone.

Migration velocity analysis: Accounting for the effects of lateral velocity variations

The most common migration velocity analysis algorithms, iterative profile migration, focusing analysis, and stack power analysis are based on restrictive subsurface assumptions that may cause the methods to break down in the presence of lateral velocity variations. Typically, the subsurface is assumed to have either a constant velocity, or at most, a depth variable velocity. Recent innovations include the use of traveltime inversion philosophy to invert migration measurements of the curvature as a function of offset exhibited by an event following migration. Traveltime inversion makes few assumptions regarding the subsurface, but is a rather unstable process. Thus, an important question is, “Under what conditions do the traditional methods break down and make the use of traveltime inversion methods mandatory?” Marrying the common migration analysis with tomography results in a set of equations that, while useful for generating updates from migration measurements, are too complex for answering the above question. However, by restricting the subsurface to low relief structures and assuming small angle wave propagation, these updating equations can be approximated by forms that are identical to the traditional updating equations, except for a factor that is dependent upon the magnitude, position, and spatial wavelength of potential lateral velocity variations. These simpler equations indicate that even relatively long wavelength anomalies can cause updates to point in the wrong direction and iterative procedures to diverge, and under certain conditions, cause the accuracy of these updates to decrease dramatically.

A Practical Approach to One-Pass 3-D Depth Migration

Accurate processing of 3-D seismic data sets is essential to meet the objectives of today's explorationist. Migration is a process that is particularly important in 3-D seismic processing. Originally, 3-D migration was performed in two separate 2-D passes but this suffered from the inaccuracies of a constant velocity assumption. The splitting algorithm is now a popular one-pass approach to 3-D migration. One-pass migration algorithms avoid the assumption of constant velocities and more correctly position the seismic energy. A more flexible approach to one-pass 3-D migration is known as k partitioning. This algorithm allows for a choice of currently available 2-D migrations to be used to perform an accurate one-pass 3-D migration. The Ky partitioning approach is also more efficient. When lateral variations in velocity become significant, a depth migration algorithm will more accurately image the data than the standard time migration algorithm. The splitting algorithm is converted into a depth migration by including the thin lens term. For the Ky partitioning algorithm this is achieved by the use of dilation. Dilation consists of small dynamic time shifts applied to the 3-D data before migration to compensate for the existence of lateral velocity variations. Application of both splitting and Ky-dilation one-pass 3-D migration algorithms to a synthetic data set show that the proposed dilation factors can produce sharp migrated images in the presence of lateral velocity variations. In this case the Ky-dilation results exhibit superior clarity compared with the image obtained from the splitting approach. The extra flexibility of the Ky-dilation approach allowed the use of a high-dip time migration algorithm which avoided the frequency dispersion effects that degrade the image of the finite-difference splitting migration. An improved image has been obtained for less computational effort.

Wave‐equation migration in the presence of lateral velocity variations

An algorithm for three‐dimensional wave equation migration in the presence of lateral velocity gradients is developed. The algorithm is based geophysically on the exploding reflector model and mathematically on integral solutions to the scalar wave equation. In addition, the velocity is assumed to vary slowly over a seismic wavelength. The primary application of the algorithm is to stacked sections. However, if the velocity is a function of depth only, the algorithm may be used to migrate and/or image common‐depth‐point gathers.