WEIGHTED STACKING FOR ROCK PROPERTY ESTIMATION AND DETECTION OF GAS*

Amplitude versus offset concepts can be used to generate weighted stacking schemes (here called geo-stack) which can be used in an otherwise standard seismic data processing sequence to display information about rock properties. The Zoeppritz equations can be simplified and several different approximations appear in the literature. They describe the variation of P-wave reflection coefficients with the angle of incidence of a P-wave as a function of the P-wave velocities, the S-wave velocities and the densities above and below an interface. Using a smooth, representative interval velocity model (from boreholes or velocity analyses) and assuming no dip, the angle of incidence can be found as a function of time and offset by iterative ray tracing. In particular, the angle of incidence can be computed for each sample in a normal moveout corrected CMP gather. The approximated Zoeppritz equation can then be fitted to the amplitudes of all the traces at each time sample of the gather, and certain rock properties can be estimated. The estimation of the rock properties is achieved by the application of time- and offset-variant weights to the data samples before stacking. The properties which can be displayed by geo-stack are: P-wave reflectivity (or true zero-offset reflectivity), S-wave reflectivity, and the reflectivity of P-wave velocity divided by S-wave velocity (or ‘pseudo-Poisson's ratio reflectivity’). If assumptions are made about the relation between P-wave velocity and S-wave velocity for water-bearing clastic silicate rocks, then it is possible to create a display which highlights the presence of gas.     

A SIMPLE EFFICIENT METHOD OF DIP-MOVEOUT CORRECTION1

Much of the success of modern seismic data processing derives from the use of the stacking process. Unfortunately, as is well known, conventional normal moveout correction (NMO) introduces mispositioning of data, and hence mis-stacking, when dip is present. Dip moveout correction (DMO) is a technique that converts non-zero-offset seismic data after NMO to true zero-offset locations and reflection times, irrespective of dip. The combination of NMO and DMO followed by post-stack time migration is equivalent to, but can be implemented much more efficiently than, full time migration before stack. In this paper we consider the frequency-wavenumber DMO algorithm developed by Hale. Our analysis centres on the result that, for a given dip, the combination of NMO at migration velocity and DMO is equivalent to NMO at the appropriate, dip-dependent, stacking velocity. This perspective on DMO leads to computationally efficient methods for applying Hale DMO and also provides interesting insights on the nature of both DMO and conventional stacking.     

VERTICAL HETEROGENEITY AND MOVEOUT IN THE ONE-DIMENSIONAL MEDIUM*

Since the important contributions of Dürbaum and Dix, 30 years ago, velocity profile estimation procedures on horizontally layered and vertically heterogeneous media from seismic probing data have been based largely on hyperbolic moveout models and RMS and stacking velocity concepts. Re-examination of the fundamentals reveals that quantitative velocity heterogeneity and canonical valocity profiles have been implicit factors for moveout modelling and for profile inversion in the use of the Dix procedure. Heterogeneity h is the ratio (and v_RMS the geometric or harmonic mean) of the path-average and time-average velocities for a raypath or, in a more restricted sense, for the normal ray belonging to a velocity profile. The canonical profile for a given velocity profile or profile segment is a moveout-equivalent monotonically increasing ramp-like profile. The ramp or constant gradient in depth is the simplest velocity profile approximator which can explicitly accommodate velocity heterogeneity. A ramp model structure is detailed which facilitates moveout simulation and model parameter estimation, and the parametric effects are explored. The horizontal offset range is quantified for which this model can give good moveout approximations.     

Generalized Dix equation and analytic treatment of normal-moveout velocity for anisotropic media*

Despite the complexity of wave propagation in anisotropic media, reflection moveout on conventional common-midpoint (CMP) spreads is usually well described by the normal-moveout (NMO) velocity defined in the zero-offset limit. In their recent work, Grechka and Tsvankin showed that the azimuthal variation of NMO velocity around a fixed CMP location generally has an elliptical form (i.e. plotting the NMO velocity in each azimuthal direction produces an ellipse) and is determined by the spatial derivatives of the slowness vector evaluated at the CMP location. This formalism is used here to develop exact solutions for the NMO velocity in anisotropic media of arbitrary symmetry. For the model of a single homogeneous layer above a dipping reflector, we obtain an explicit NMO expression valid for all pure modes and any orientation of the CMP line with respect to the reflector strike. The contribution of anisotropy to NMO velocity is contained in the slowness components of the zero-offset ray (along with the derivatives of the vertical slowness with respect to the horizontal slownesses) — quantities that can be found in a straightforward way from the Christoffel equation. If the medium above a dipping reflector is horizontally stratified, the effective NMO velocity is determined through a Dix-type average of the matrices responsible for the ‘interval’ NMO ellipses in the individual layers. This generalized Dix equation provides an analytic basis for moveout inversion in vertically inhomogeneous, arbitrarily anisotropic media. For models with a throughgoing vertical symmetry plane (i.e. if the dip plane of the reflector coincides with a symmetry plane of the overburden), the semi-axes of the NMO ellipse are found by the more conventional rms averaging of the interval NMO velocities in the dip and strike directions. Modelling of normal moveout in general heterogeneous anisotropic media requires dynamic ray tracing of only one (zero-offset) ray. Remarkably, the expressions for geometrical spreading along the zero-offset ray contain all the components necessary to build the NMO ellipse. This method is orders of magnitude faster than multi-azimuth, multi-offset ray tracing and, therefore, can be used efficiently in traveltime inversion and in devising fast dip-moveout (DMO) processing algorithms for anisotropic media. This technique becomes especially efficient if the model consists of homogeneous layers or blocks separated by smooth interfaces. The high accuracy of our NMO expressions is illustrated by comparison with ray-traced reflection traveltimes in piecewise-homogeneous, azimuthally anisotropic models. We also apply the generalized Dix equation to field data collected over a fractured reservoir and show that P-wave moveout can be used to find the depth-dependent fracture orientation and to evaluate the magnitude of azimuthal anisotropy.     

Migration to multiple offset and velocity analysis1

The stacking velocity best characterizes the normal moveout curves in a common-mid-point gather, while the migration velocity characterizes the diffraction curves in a zero-offset section as well as in a common-midpoint gather. For horizontally layered media, the two velocity types coincide due to the conformance of the normal and the image ray. In the case of dipping subsurface structures, stacking velocities depend on the dip of the reflector and relate to normal rays, but with a dip-dependent lateral smear of the reflection point. After dip-moveout correction, the stacking velocities are reduced while the reflection-point smear vanishes, focusing the rays on the common reflection points. For homogeneous media the dip-moveout correction is independent of the actual velocity and can be applied as a dip-moveout correction to multiple offset before velocity analysis. Migration to multiple offset is a prestack, time-migration technique, which presents data sets which mimic high-fold, bin-centre adjusted, common-midpoint gathers. This method is independent of velocity and can migrate any 2D or 3D data set with arbitrary acquisition geometry. The gathers generated can be analysed for normal-moveout velocities using traditional methods such as the interpretation of multivelocity-function stacks. These stacks, however, are equivalent to multi-velocity-function time migrations and the derived velocities are migration velocities.     

P‐wave stacking‐velocity tomography for VTI media

A major complication caused by anisotropy in velocity analysis and imaging is the uncertainty in estimating the vertical velocity and depth scale of the model from surface data. For laterally homogeneous VTI (transversely isotropic with a vertical symmetry axis) media above the target reflector, P‐wave moveout has to be combined with other information (e.g. borehole data or converted waves) to build velocity models for depth imaging. The presence of lateral heterogeneity in the overburden creates the dependence of P‐wave reflection data on all three relevant parameters (the vertical velocity V_P0 and the Thomsen coefficients ε and δ) and, therefore, may help to determine the depth scale of the velocity field. Here, we propose a tomographic algorithm designed to invert NMO ellipses (obtained from azimuthally varying stacking velocities) and zero‐offset traveltimes of P‐waves for the parameters of homogeneous VTI layers separated by either plane dipping or curved interfaces. For plane non‐intersecting layer boundaries, the interval parameters cannot be recovered from P‐wave moveout in a unique way. Nonetheless, if the reflectors have sufficiently different azimuths, a priori knowledge of any single interval parameter makes it possible to reconstruct the whole model in depth. For example, the parameter estimation becomes unique if the subsurface layer is known to be isotropic. In the case of 2D inversion on the dip line of co‐orientated reflectors, it is necessary to specify one parameter (e.g. the vertical velocity) per layer. Despite the higher complexity of models with curved interfaces, the increased angle coverage of reflected rays helps to resolve the trade‐offs between the medium parameters. Singular value decomposition (SVD) shows that in the presence of sufficient interface curvature all parameters needed for anisotropic depth processing can be obtained solely from conventional‐spread P‐wave moveout. By performing tests on noise‐contaminated data we demonstrate that the tomographic inversion procedure reconstructs both the interfaces and the VTI parameters with high accuracy. Both SVD analysis and moveout inversion are implemented using an efficient modelling technique based on the theory of NMO‐velocity surfaces generalized for wave propagation through curved interfaces.     

Dense 3D residual moveout analysis as a tool for HTI parameter estimation

Three‐dimensional residual moveout analysis is the basic step in velocity model refinement. The analysis is generally carried out using horizontal and/or vertical semblances defined on a sparse set of in‐lines or cross‐lines with densely sampled source–receiver offsets. An alternative approach, which we call dense residual moveout analysis (DRMA), is to use all the bins of a three‐dimensional survey but sparsely sampled offsets. The proposed technique is very fast and provides unbiased and statistically efficient estimates of the residual moveout. Indeed, for the sparsest possible offset distribution, when only near‐ and far‐angle stacks are used, the variance of the residual moveout estimate is only 1.4 times larger than the variance of the least‐squares estimate obtained using all offsets. The high performance of DRMA makes it a useful tool for many applications, of which azimuthal velocity analysis is considered here. For a horizontal transverse isotropy (HTI) model, a deterministic procedure is proposed to define, at every point of residual moveout estimation, the azimuthal angle of the HTI axis of symmetry, the Thomsen anisotropy coefficients, and the interval (or root‐mean‐square) velocities in both the HTI isotropy and symmetry planes. The procedure is not restricted by DRMA assumptions; for example, it is also applicable to semblance‐based residual moveout estimates. The high resolution of the technique is illustrated by azimuthal velocity analysis over an oilfield in West Siberia.     

Reverse time migration of CMP-gathers an effective tool for the determination of interval velocities

One of the most important steps in the conventional processing of reflection seismic data is common midpoint (CMP) stacking. However, this step has considerable deficiencies. For instance the reflection or diffraction time curves used for normal moveout corrections must be hyperbolae. Furthermore, undesirable frequency changes by stretching are produced on account of the dependence of the normal moveout corrections on reflection times. Still other drawbacks of conventional CMP stacking could be listed.One possibility to avoid these disadvantages is to replace conventional CMP stacking by a process of migration to be discussed in this paper. For this purpose the Sherwood-Loewenthal model of the exploding reflector has to be extended to an exploding point model with symmetry to the lineP _EX M whereP _EX is the exploding point, alias common reflection point, andM the common midpoint of receiver and source pairs.Kirchhoff summation is that kind of migration which is practically identical with conventional CMP stacking with the exception that Kirchhoff summation provides more than one resulting trace.In this paper reverse time migration (RTM) was adopted as a tool to replace conventional CMP stacking. This method has the merit that it uses the full wave equation and that a direct depth migration is obtained, the velocityv can be any function of the local coordinatesx, y, z. Since the quality of the reverse time migration is highly dependent on the correct choice of interval velocities such interval velocities can be determined stepwise from layer to layer, and there is no need to compute interval velocities from normal moveout velocities by sophisticated mathematics or time consuming modelling. It will be shown that curve velocity interfaces do not impair the correct determination of interval velocities and that more precise velocity values are obtained by avoiding or restricting muting due to non-hyperbolic normal moveout curves.Finally it is discussed how in the case of complicated structures the reverse time migration of CMP gathers can be modified in such a manner that the combination of all reverse time migrated CMP gathers yields a correct depth migrated section. This presupposes, however, a preliminary data processing and interpretation.     

Stacking velocities in the presence of overburden velocity anomalies

Lateral velocity changes (velocity anomalies) in the overburden may cause significant oscillations in normal moveout velocities. Explicit analytical moveout formulas are presented and provide a direct explanation of these lateral fluctuations and other phenomena for a subsurface with gentle deep structures and shallow overburden anomalies. The analytical conditions for this have been derived for a depth-velocity model with gentle structures with dips not exceeding 12°. The influence of lateral interval velocity changes and curvilinear overburden velocity boundaries can be estimated and analysed using these formulas. An analytical approach to normal moveout velocity analysis in a laterally inhomogeneous medium provides an understanding of the connection between lateral interval velocity changes and normal moveout velocities. In the presence of uncorrected shallow velocity anomalies, the difference between root-mean-square and stacking velocity can be arbitrarily large to the extent of reversing the normal moveout function around normal incidence traveltimes. The main reason for anomalous stacking velocity behaviour is non-linear lateral variations in the shallow overburden interval velocities or the velocity boundaries.
A special technique has been developed to determine and remove shallow velocity anomaly effects. This technique includes automatic continuous velocity picking, an inversion method for the determination of shallow velocity anomalies, improving the depth-velocity model by an optimization approach to traveltime inversion (layered reflection tomography) and shallow velocity anomaly replacement. Model and field data examples are used to illustrate this technique.     

一种倾角时差校正和叠前成象方法

本文从测量射线参数出发进行反向射线追踪,导出倾角时差校正（DMO）的公式。经过DMO后,可以从一组等炮检距剖面得出共分角线点道集。用于对这些道集进行叠加的速度值与界面倾角无关。对经过DMO的资料的等时切片进行叠前成象（PSI）,就可以把分布在圆上的绕射能量沿圆弧加起来,并放在圆弧上对应于最大炮检距的位置。经过这两种处理,再应用标准的速度分析和叠加方法,就可得出偏移后的剖面。这两种处理均与速度无关。最后用物理模型试验说明了DMO和PSI的效果是好的。     

STACKING OF P-SV SEISMIC REFLECTION DATA USING DIP MOVEOUT1

The moveout of P-SV mode-converted seismic reflection events in a common-midpoint gather is non-hyperbolic. This is true even if the medium has constant P- and SV-wave velocities. Furthermore, reflection-point smear occurs even along horizontal reflectors. These effects reduce the resolution of the zero-offset stack. In such a medium, the generalization of the dip moveout transformation to P-SV data can be calculated analytically. The resulting P-SV dip moveout operators solve the problem of reflection-point smear, and image any reflector regardless of dip or depth. The viability of this technique is demonstrated on synthetic and field data.     

Seismic imaging of deep hydrocarbon reservoirs

This paper examines those aspects of reflection seismology which require special consideration when imaging deeper hydrocarbon reservoirs, including the constraints imposed by vertical resolution, lateral resolution, and velocity analysis. We derive quantitative expressions relating the uncertainties in stacking velocities and in interval velocities derived from stacking velocities to acquisition parameters, as well as expressions for the lateral resolution which can theoretically be achieved for migrated seismic images. This analysis shows that the most significant limitations of seismic imaging at depth involve the finite lateral resolution of the seismic method, and the proper lateral positioning of seismic images. These difficulties are overcome in large measure through the proper migration of a seismic dataset, which becomes more critical as deeper horizons are imaged. If these horizons are suspected of having significant 3-D structure, a strong argument may be made for acquiring a 3-D seismic survey over the prospect. Migration of this dataset will then generate an image of the subsurface with good lateral resolution in both the X and Y directions.     

DIFFERENT METHODS OF TIME-DEPTH CONVERSION WITH AND WITHOUT MIGRATION*

The increasing use of velocity analysis programs in seismic processing in addition to direct application to normal move out corrections often makes it possible to study the variation of the average velocity versus time and distance. Usually, it is even feasible to compute interval velocities with a good accuracy. We intend to use these velocities to convert the usual time domain displays to the depth domain. An exact conversion requires the knowledge of all interval velocities and all the velocity interfaces. However, when the dips are small, the conversion can be done without considering the dip of all these interfaces, and in some cases migration can even be completely neglected. Three different programs will be described and their use discussed in view of the type of problems to which they are applied. Practical examples of the various methods will be presented.     

COHERENCY WEIGHTING—AN EFFECTIVE APPROACH TO THE SUPPRESSION OF LONG LEG MULTIPLES*

Long leg multiples can be suppressed by a method which provides an alternative to weighted common-depth-point stacking and multichannel stacking filtering. The suppression is achieved by coherency weighting whereby the time-dependent weighting factor decreases as the semblance of the multiple reflections increases. The algorithm of the method is described. Its efficiency is discussed in relation to the input data and results of its application to marine seismic data are presented. For practical application, the stacking velocity of the multiples has to be known. As the process is based on stacking velocities, different types of multiples can be handled, for instance water-bottom multiples or internal multiples. The parameter analysis shows that the degree of multiple suppression can easily be controlled by adapting the parameters of the procedure to the field conditions. During the suppression of multiples, the primaries are saved according to the moveout differences between the two. The non-linear behaviour of the process causes signal suppression and distortion effects, which have to be corrected by AGC normalization and low-pass filtering. Among the various applications available, only the suppression of long leg water-bottom multiples is treated here. The results show that their suppression on the basis of moveout differences is efficient even when standard length streamers are used in regions with water depth of up to 1500 m and more, if the stacking velocity of the primaries is about 10 to 20% higher than that of the multiples. Even if those parts of the primaries which are masked by the multiples are suppressed in the individual common-depth-point gathers by the procedure, the remaining primaries in the AGV stacked section are largely uncovered by the multiple suppression.     

Common-reflection-surface stack — a real data example

The simulation of a zero-offset (ZO) stack section from multi-coverage reflection data is a standard imaging method in seismic processing. It significantly reduces the amount of data and increases the signal-to-noise ratio due to constructive interference of correlated events. Conventional imaging methods, e.g., normal moveout (NMO)/dip moveout (DMO)/stack or pre-stack migration, require a sufficiently accurate macro-velocity model to yield appropriate results, whereas the recently introduced common-reflection-surface stack does not depend on a macro-velocity model. For two-dimensional seismic acquisition, its stacking operator depends on three wavefield attributes and approximates the kinematic multi-coverage reflection response of curved interfaces in laterally inhomogeneous media. The common-reflection-surface stack moveout formula defines a stacking surface for each particular sample in the ZO section to be simulated. The stacking surfaces that fit best to actual events in the multi-coverage data set are determined by means of coherency analysis. In this way, we obtain a coherency section and a section of each of the three wavefield attributes defining the stacking operator. These wavefield attributes characterize the curved interfaces and, thus, can be used for a subsequent inversion. In this paper, we focus on an application to a real land data set acquired over a salt dome. We propose three separate one-parametric search and coherency analyses to determine initial common-reflection-surface stack parameters. Optionally, a subsequent optimization algorithm can be performed to refine these initial parameters. The simulated ZO section obtained by the common-reflection-surface stack is compared to the result of a conventional NMO/DMO/stack processing sequence. We observe an increased signal-to-noise ratio and an improved continuity along the events for our proposed method — without loss of lateral resolution.     

OFFSET CONTINUATION OF SEISMIC SECTIONS*

Geometrical acoustic and wave theory lead to a second-order partial differential equation that links seismic sections with different offsets. In this equation a time-shift term appears that corresponds to normal moveout; a second term, dependent on offset and time only, corrects the moveout of dipping events. The zero-offset stacked section can thus be obtained by continuing the section with maximum offset towards zero, and stacking along the way the other common-offset sections. Without the correction for dip moveout, the spatial resolution of the section is noticeably impaired, thus limiting the advantages that could be obtained with expensive migration procedures. Trade-offs exist between multiplicity of coverage, spatial resolution, and signal-to-noise; in some cases the spatial resolution on the surface can be doubled and the aliasing noise averaged out. Velocity analyses carried out on data continued to zero offset show a better resolution and improved discrimination against multiples. For instance, sea-floor multiples always appear at water velocity, so that their removal is simplified. This offset continuation can be carried out either in the time-space domain or in the time-wave number domain. The methods are applied both to synthetic and real data.     

COMPUTATION OF THE NORMAL MOVEOUT VELOCITY IN 3D LATERALLY INHOMOGENEOUS MEDIA WITH CURVED INTERFACES*

For a 3D velocity model of curved first order interfaces and layer velocities which are arbitrary smooth functions of the space coordinates, the normal moveout (NMO)-velocity can be computed by numerically integrating a system of first order ordinary differential equations for a hypothetical wavefront that originates at the normal incidence point of the normal ray and moves up along the ray to the common mid-point of the common datum point (CDP) profile.     

Kinematic parameters of pure‐ and converted‐mode waves in elastic orthorhombic media

I derive the kinematic properties of single‐mode P, S1, and S2 waves as well as converted PS1, PS2, and S1S2 waves in elastic orthorhombic media including vertical velocity, two normal moveout velocities defined in vertical symmetry planes, and three anelliptic parameters (two of them are defined in vertical symmetry plane and one parameter is the cross‐term one). I show that the azimuthal dependence of normal moveout velocity and anellipticity is different in phase and group domains. The effects on‐vertical‐axis singularity and on‐vertical‐axis triplication are considered for pure‐mode S1 and S2 waves and converted‐mode S1S2 waves. The conditions and properties of on‐vertical‐axis triplication are defined in terms of kinematic parameters. The results are illustrated in four homogeneous orthorhombic models and one multilayered orthorhombic model with no variation in azimuthal orientation for all the layers.     

Characterization of dipping fractures in a transverselyisotropic background

Although it is believed that natural fracture sets predominantly have near‐vertical orientation, oblique stresses and some other mechanisms may tilt fractures away from the vertical. Here, we examine an effective medium produced by a single system of obliquely dipping rotationally invariant fractures embedded in a transversely isotropic with a vertical symmetry axis (VTI) background rock. This model is monoclinic with a vertical symmetry plane that coincides with the dip plane of the fractures. Multicomponent seismic data acquired over such a medium possess several distinct features that make it possible to estimate the fracture orientation. For example, the vertically propagating fast shear wave (and the fast converted PS‐wave) is typically polarized in the direction of the fracture strike. The normal‐moveout (NMO) ellipses of horizontal reflection events are co‐orientated with the dip and strike directions of the fractures, which provides an independent estimate of the fracture azimuth. However, the polarization vector of the slow shear wave at vertical incidence does not lie in the horizontal plane – an unusual phenomenon that can be used to evaluate fracture dip. Also, for oblique fractures the shear‐wave splitting coefficient at vertical incidence becomes dependent on fracture infill (saturation). A complete medium‐characterization procedure includes estimating the fracture compliances and orientation (dip and azimuth), as well as the Thomsen parameters of the VTI background. We demonstrate that both the fracture and background parameters can be obtained from multicomponent wide‐azimuth data using the vertical velocities and NMO ellipses of PP‐waves and two split SS‐waves (or the traveltimes of PS‐waves) reflected from horizontal interfaces. Numerical tests corroborate the accuracy and stability of the inversion algorithm based on the exact expressions for the vertical and NMO velocities.