Zvi Koren

Wavelengths of Earth structures that can be resolved from seismic reflection data

The aim of inverting seismic waveforms is to obtain the “best” earth model. The best model is defined as the one producing seismograms that best match (usually under a least‐squares criterion) those recorded. Our approach is nonlinear in the sense that we synthesize seismograms without using any linearization of the elastic wave equation. Since we use rather complete data sets without any spatial aliasing, we do not have the problem of secondary minima (Tarantola, 1986). Nevertheless, our gradient methods fail to converge if the starting earth model is far from the true earth (Mora, 1987; Kolb et al., 1986; Pica et al., 1989).

Monte Carlo estimation and resolution analysis of seismic background velocities

The complete solution to an inverse problem, including information on accuracy and resolution, is given by the a posteriori probability density in the model space. By running a modified simulated annealing, samples from the model space can be drawn in such a way that their frequencies of occurrence approximate their a posteriori likelihoods. Using this method, maximum likelihood estimation and uncertainty analysis of seismic background velocity models are performed on multioffset seismic data. The misfit between observed and synthetic waveforms within the time windows along computed multioffset travel times, is used as an objective function for the simulated annealing approach. The real medium is modeled as a series of layers separated by curved interfaces. Lateral velocity variations within the layers are determined by interpolation from specified values at a number of sampling points. The input data consists of multioffset seismic data. Additionally, zero-offset times are used to migrate the reflectors in time to the depth domain. The multioffset times are calculated by an efficient ray-tracing algorithm which allows inversion of a large number of seismograms. The a posteriori probability density for this problem is highly multidimensional and highly multimodal. Therefore, the information contained in this distribution cannot be adequately represented by standard deviations and covariances. However, by sequentially displaying a large number of images, computed from the a posteriori background velocity samples and the data, it is possible to convey to the spectator a better understanding of what information we really have on the subsurface.

Full-Azimuth Angle Domain Imaging

This work presents a new seismic imaging system for generating and extracting high-resolution information about subsurface angle dependent reflectivity, with simultaneous emphasis on both continuous structural surfaces and discontinuous objects, such as faults and small-scale fractures. The system enables full-azimuth, angledependent seismic imaging using reflection data recorded through seismic acquisition surveys, especially wideazimuth and long offset data. Geometrical attributes, such as dip-azimuth and continuity of the local reflecting surfaces, can be automatically extracted directly from the full-azimuth angle gathers. Azimuthal anisotropy can be detected, leading to an accurate anisotropy model representation.

Target-Oriented Common Reflection Angle Migration

Summary A 2-D and 3-D ray-based migration/inversion approach for the construction of common image gathers (CIG) in the reflection angle domain is presented. We show that amplitudes and phases of the reflected events are preserved for a wide range of angles even in complex areas with multi-arrivals. The method can be used for detailed velocity-model determination and for accurate amplitude variation with angle (AVA) analysis in such areas. Our method is a target-oriented approach, based on shooting rays from the image points up to the surface. The migration aperture and density of rays per solid dip angle at the image points is chosen with the condition that we obtain optimal reconstruction of the migrated events, avoiding migration operator aliasing. However, using the whole migration aperture might be very time consuming, especially for steep angles, which require small angle step increments. We therefore present an implementation of a model-driven aperture migration, which makes the migration feasible and relatively fast even for large-scale complex 3-D models. The migration aperture is defined from information about the local directivity of the main reflectors, obtained from interpreted horizons. The implementation of the method for complex geological structures is demonstrated with the 2-D Marmousi dataset and with the 3D SEG/EAGE salt model.

Anisotropic local tomography

Local tomography is interactive, ray-based, residual-interval-parameter analysis for updating background anisotropic velocity parameters. The method operates directly on image gathers generated by anisotropic curved-ray Kirchhoff time migration. A locally 1D, spatially varying, vertical transversely isotropic model is assumed. The background anisotropy parameters are the instantaneous (interval) vertical compression velocity VP and the two Thomsen anisotropy parameters, δ and e . The interval velocity δ is updated from short-offset reflection events, and e is updated from available long-offset data. The medium parameters are updated from the top down both vertically and by layers, one parameter at a time. The picked residual-anisotropy parameters correspond to the residual-moveout (RMO) curves that best fit the migrated reflection events. The method is based on splitting the contribution to the computed RMO at a given point into two parts: from overburden residual parameters and from the actual picked res...

Normal moveout velocity for pure‐mode and converted waves in layered orthorhombic medium

We study the azimuthally dependent hyperbolic moveout approximation for small nangles (or offsets) for quasi-compressional, quasi-shear, and converted waves in onedimensional nmulti-layer orthorhombic media. The vertical orthorhombic axis is the nsame for all layers, but the azimuthal orientation of the horizontal orthorhombic naxes at each layer may be different. By starting with the known equation for normal nmoveout velocity with respect to the surface-offset azimuth and applying our nderived relationship between the surface-offset azimuth and phase-velocity azimuth, nwe obtain the normal moveout velocity versus the phase-velocity azimuth. As the surface noffset/azimuth moveout dependence is required for analysing azimuthally dependent nmoveout parameters directly from time-domain rich azimuth gathers, our phase nangle/azimuth formulas are required for analysing azimuthally dependent residual nmoveout along the migrated local-angle-domain common image gathers. The angle nand azimuth parameters of the local-angle-domain gathers represent the opening angle nbetween the incidence and reflection slowness vectors and the azimuth of the nphase velocity ψphs at the image points in the specular direction. Our derivation of nthe effective velocity parameters for a multi-layer structure is based on the fact that, nfor a one-dimensional model assumption, the horizontal slowness ph and the azimuth nof the phase velocity ψphs remain constant along the entire ray (wave) path. We introduce na special set of auxiliary parameters that allow us to establish equivalent neffective model parameters in a simple summation manner. We then transform this nset of parameters into three widely used effective parameters: fast and slow normal nmoveout velocities and azimuth of the slow one. For completeness, we show that these nthree effective normal moveout velocity parameters can be equivalently obtained in nboth surface-offset azimuth and phase-velocity azimuth domains.

Specular/diffraction imaging by full azimuth subsurface angle domain decomposition

This work presents a new seismic imaging system for generating amplitude preserved, three-dimensional directional gathers. The proposed method is based on directional angle decomposition that enables the implementation of both specular and diffraction imaging in real 3D isotropic/anisotropic geological models, leading to simultaneous emphasis on both continuous structural surfaces and discontinuous objects, such as faults and small-scale fractures. Structural attributes at each subsurface point, e.g., dip, azimuth and continuity, can be derived directly from the directional angle gathers. The proposed approach is most effective for imaging and analysis below complex structures, such as subsalt and subbasalt, high-velocity carbonate rocks, shallow velocity anomalies, and others.

Fourth-order normal moveout velocity in elastic layered orthorhombic media — Part 1: Slowness-azimuth domain

Considering all types of pure-mode and converted waves, we derive the azimuthally dependent, fourth-order normal moveout (NMO) velocity functions, and hence the corresponding effective anellipticity functions, for horizontally layered orthorhombic media. We emphasize that this paper does not suggest a new nonhyperbolic traveltime approximation; rather, it provides exact expressions of the NMO series coefficients, computed for normal-incidence rays, which can then be further used within known azimuthally dependent traveltime approximations for short to moderate offsets. We do not assume weak anisotropy or acoustic approximation for P-waves. At each layer, the elastic parameters, thickness, and azimuth of the orthorhombic vertical symmetry planes are considered to be different. We distinguish between two different azimuths: slowness azimuth (part 1 of this paper) and offset azimuth (part 2 of this paper). In part 1, the slownessazimuth domain NMO is approximated as a series of either infinitesimal horizontal slowness (slowness-azimuth/slowness domain) or infinitesimal offsets (slowness-azimuth/offset domain). Similarly, in part 2, we distinguish between two offset-azimuth domains: offset-azimuth/slowness and offset-azimuth/offset. Note that the azimuthally dependent NMO velocity functions of each of the four cases are different. The validity of the method is tested by introducing our derived azimuthally dependent, fourth-order effective anellipticity, into the well-known azimuthally dependent, asymptotic nonhyperbolic traveltime approximation, in which we compare the traveltime approximation versus exact numerical ray tracing for short to moderate offsets. It is clearly shown that for these types of azimuthally anisotropic layered models, the fourth-order terms are essential even for relatively small horizontal-slowness values or short offsets.

Optimum time‐to‐depth conversion

Time-to-depth conversion is usually accomplished by converting zero-offset traveltimes, interpreted from a stacked section, to depth using a known velocity field. Time-to-depth conversion is formulated as an iterative procedure producing a depth model which minimizes the differences between zero-offset picked traveltimes and times derived by normal-incidence ray tracing through the model. The input data consist of traveltimes picked from unmigrated stacked sections, as well as assumed interval velocities in each layer. The method can be applied to models with discontinuities such as pinchouts and faults. To illustrate the method, synthetic and field data examples are included.

Traveltime approximation in vertical transversely isotropic layered media

The well-known asymptotic fractional four-parameter traveltime approximation and nthe five-parameter generalised traveltime approximation in stratified multi-layer ntransversely isotropic elastic media with a vertical axis of symmetry have been widely nused for pure-mode and converted waves. The first three parameters of these traveltime nexpansions are zero-offset traveltime, normal moveout velocity, and quartic ncoefficient, ensuring high accuracy of traveltimes at short offsets. The additional parameter nwithin the four-parameter approximation is an effective horizontal velocity naccounting for large offsets, which is important to avoid traveltime divergence at nlarge offsets. The two additional parameters in the above-mentioned five-parameter napproximation ensure higher accuracy up to a given large finite offset with an exact nmatch at this offset. In this paper, we propose two alternative five-parameter traveltime napproximations, which can be considered extensions of the four-parameter napproximation and an alternative to the five-parameter approximation previously nmentioned. The first three short-offset parameters are the same as before, but the two nadditional long-offset parameters are different and have specific physical meaning. nOne of them describes the propagation in the high-velocity layer of the overburden n(nearly horizontal propagation in the case of very large offsets), and the other ncharacterises the intercept time corresponding to the critical slowness that includes ncontributions of the lower velocity layers only. Unlike the above-mentioned approximations, nboth of the proposed traveltime approximations converge to the theoretical n(asymptotic) linear traveltime at the limit case of very large (“infinite”) offsets. Their naccuracy for moderate to very large offsets, for quasi-compressional waves, converted nwaves, and shear waves polarised in the horizontal plane, is extremely high in cases nwhere the overburden model contains at least one layer with a dominant higher velocity ncompared with the other layers. We consider the implementation of the proposed ntraveltime approximations in all classes of problems in which the above-mentioned napproximations are used, such as reflection and diffraction analysis and imaging.