Azimuthally dependent anisotropic velocity model update

Summary Numerical and physical modelling studies have been used to investigate the accuracy and limitations of the small-offset P-wave NMO velocity estimation in transversely isotropic media. An expression for the P-wave small-offset NMO velocity for a single horizontal reflector in a vertically transversely isotropic layer has been previously obtained: V nmo =+ αδ 0 12 where αο and δ respectively represent the vertical P-wave velocity and the P-wave critical anisotropy at oblique angles. The above equation is presented as being exact and valid for any degree of anisotropy and has been used in estimating the small-off set NMO velocity in the offshore Zaire seismic data acquired by Unocal in West Africa. The accuracy of this expression is subject t o practical testing in this paper. A one-layer horizontal model was used to carry out the numerical modelling studies. A computer program which calculates Pwave NMO velocities in transversely isotropic media was used. This program uses the exact equations for the phase and ray velocities in generating velocity functions to be used in ray tracing; it is known to give accurate results. Using the publishe d elastic parameters for anisotropic rocks, the small-offset NMO velocities are computed and results are compared with values obtained using the above expression. The degree of anisotropy varied from very weak to strong and contrasting velocity function s (positive and negative δ) were used in numerical modelling simulations. Phenolite materials with contrasting velocity functions were used to simulate transversely isotropic media with vertical axis of symmetry in physical modelling tests. Experimental reflection data were collected and velocity analysis was conducted for small-offset tr aces, and the NMO velocity determined by the maximum stack response. The elastic parameters αο and δ of the Phenolite materials were recovered by an anisotropic inversion technique. These parameters were used to estimate the small-offset P-wave NMO velocity in Phenolite with the above expression and also with our computer program. The NMO velocities obtained from these two methods were compared to that obtained from velocity analysis of the experimental seismic data. These comparisons enabled the validity of the above expressio n to yield accurate P-wave NMO velocity, in the limit of small offset, to be tested. Numerical and physical modelling results indicate that the accuracy of the small-offset P-wave NMO equation depends on the nature and degree of the anisotropy encountered in a given area, and is not valid for all degrees of P-wave anisotropy. The sma lloffset NMO (stacking) velocity from horizontal reflectors is known to influence the effects of vertical transverse isotropy on normal moveout (NMO), dip moveout (DMO) corrections and time migration. Determination of the correct small-offset P-wave NMO velocity is essential for obtaining accurate seismic images, and subsequently correct interpretations.

The paper presents estimates of the S-wave velocity and the crack density at which fractured reservoirs begin to play an important role in oil exploration. Transverse isotropy with a horizontal axis of symmetry (HTI) is the simplest azimuthally anisotropic model used to describe fractured reservoirs that contain parallel vertical cracks. A double profile concept is used to develop an equation for the P-S wave normal-moveout (NMO) velocity. The azimuthal NMO velocities of the P- and P-S waves can then be used to estimate the velocities of the S-waves and Thomsen's coefficient, γ. For multilayered media, a recursive equation is developed for the NMO velocity in each layer. The numerical results indicate that the S-wave NMO velocity can be accurately estimated using the P- and P-S wave NMO velocities in HTI media. An important parameter of fracture systems that can be measured from seismic data is the crack density which can be estimated using the NMO velocities of the P- and S-waves from horizontal reflectors. Therefore, fractures can be completely characterized by the joint inversion of P-waves and converted P-S waves in HTI media.

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

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.

Anisotropic P-wave velocity analysis and seismic imaging in onshore Kutch sedimentary basin of India

Seismic velocity modeling is a crucial step in seismic processing that enables the use of velocity information from both seismic and wells to map the depth and thickness of subsurface layers interpreted from seismic images. The velocity can be obtained in the form of normal moveout (NMO) velocity or by an inversion (optimization) process such as in full-waveform inversion (FWI). These methods have several limitations. These limitations include enormous time consumption in the case of NMO due to manual and heavy human involvement in the picking. As an optimization problem, it incurs high cost and suffers from nonlinearity issues. Researchers have proposed various machine learning (ML) techniques including unsupervised, supervised, and semi-supervised learning methods to model the velocity more efficiently. The focus of the studies is mostly to automate the NMO velocity picking, improve the convergence in FWI, and apply FWI using ML directly from the data. In the purview of the digital transformation roadmap of the petroleum industry, this paper presents a chronologic review of these studies, appraises the progress made so far, and concludes with a set of recommendations to overcome the prevailing challenges through the implementation of more advanced ML methodologies. We hope that this work will benefit experts, young professionals, and ML enthusiasts to help push forward their research efforts to achieving complete automation of the NMO velocity and further enhancing the performance of ML applications used in the FWI framework.

The main difficulty in extending seismic processing to anisotropic media is the recovery of anisotropic velocity fields from surface reflection data. We suggest carrying out velocity analysis for transversely isotropic (TI) media by inverting the dependence of P-wave moveout velocities on the ray parameter. The inversion technique is based on the exact analytic equation for the normal-moveout (NMO) velocity for dipping reflectors in anisotropic media.We show that P-wave NMO velocity for dipping reflectors in homogeneous TI media with a vertical symmetry axis depends just on the zero-dip value V nmo (0) and a new effective parameter eta that reduces to the difference between Thomsen parameters epsilon and delta in the limit of weak anisotropy. Our inversion procedure makes it possible to obtain eta and reconstruct the NMO velocity as a function of ray parameter using moveout velocities for two different dips. Moreover, V nmo (0) and eta determine not only the NMO velocity, but also long-spread (nonhyperbolic) P-wave moveout for horizontal reflectors and the time-migration impulse response. This means that inversion of dip-moveout information allows one to perform all time-processing steps in TI media using only surface P-wave data. For elliptical anisotropy (epsilon = delta ), isotropic time-processing methods remain entirely valid. We show the performance of our velocity-analysis method not only on synthetic, but also on field data from offshore Africa.Accurate time-to-depth conversion, however, requires that the vertical velocity V P0 be resolved independently. Unfortunately, it cannot be done using P-wave surface moveout data alone, no matter how many dips are available. In some cases V P0 is known (e.g., from check shots or well logs); then the anisotropy parameters epsilon and delta can be found by inverting two P-wave NMO velocities corresponding to a horizontal and a dipping reflector. If no well information is available, all three parameters (V P0 , epsilon , and delta ) can be obtained by combining our inversion results with shear-wave information, such as the P-SV or SV-SV wave NMO velocities for a horizontal reflector.Generalization of the single-layer NMO equation to layered anisotropic media with a dipping reflector provides a basis for extending anisotropic velocity analysis to vertically inhomogeneous media. We demonstrate how the influence of a stratified anisotropic overburden on moveout velocity can be stripped through a Dix-type differentiation procedure.

Fluid flow in many hydrocarbon reservoirs is controlled by aligned fractures which make the medium anisotropic on the scale of seismic wavelength. Applying the linear‐slip theory, we investigate seismic signatures of the effective medium produced by a single set of ‘general’ vertical fractures embedded in a purely isotropic host rock. The generality of our fracture model means the allowance for coupling between the normal (to the fracture plane) stress and the tangential jump in displacement (and vice versa). Despite its low (triclinic) symmetry, the medium is described by just nine independent effective parameters and possesses several distinct features which help to identify the physical model and estimate the fracture compliances and background velocities. For example, the polarization vector of the vertically propagating fast shear wave S1 and the semi‐major axis of the S1‐wave normal‐moveout (NMO) ellipse from a horizontal reflector always point in the direction of the fracture strike. Moreover, for the S1‐wave both the vertical velocity and the NMO velocity along the fractures are equal to the shear‐wave velocity in the host rock. Analysis of seismic signatures in the limit of small fracture weaknesses allows us to select the input data needed for unambiguous fracture characterization. The fracture and background parameters can be estimated using the NMO ellipses from horizontal reflectors and vertical velocities of P‐waves and two split S‐waves, combined with a portion of the P‐wave slowness surface reconstructed from multi‐azimuth walkaway vertical seismic profiling (VSP) data. The stability of the parameter‐estimation procedure is verified by performing non‐linear inversion based on the exact equations.

Surface-consistent residual statics correction for land seismic data does not account for the source - receiver offset. Consequently, it requires normal moveout (NMO) corrected gathers to bring raypaths close to the normal incidence. When the NMO velocity is inaccurate or unavailable, the estimated statics suffer. Therefore, multiple passes of NMO velocity picking and residual statics estimation become essential, which are efforts and time consuming. To avoid this, we utilize a rank-based solution that is capable of estimating non-surface consistent residual statics. The method is based on the rank property of frequency slices in the midpoint-offset domain, where ideal seismic data is of low-rank nature, while data with residual statics exhibits higher rank. Accordingly, we estimate the statics that lead to the desired low-rank signal via means of low-rank approximation and cross-correlation in an iterative and multi-rank-scale approach. Since we estimate non surface-consistent statics by accounting for the offset of each trace, it is no longer required to have NMO corrected gathers. Consequently, the method does not require windowing over a noise-free section containing primaries or windowing to avoid the NMO stretch effect, which are required by conventional residual statics correction. Numerical results on simulated and field data suggest that the method has the potential of replacing existing residual statics correction techniques.

The anisotropic phenomenon in wave propagation has been widely recognized on various scales. Knowledge of anisotropic effects is essential for interpretation and processing of seismic data. The existence of anisotropy leads to the moveout of nonhyperbolic even in a homogenous layer. A transversely isotropic medium with a vertical symmetry axis (VTI) is a reasonable approximation of horizontally layered anisotropic medium. The approximation of travel time is important for reducing the residual normal moveout of layered VTI media. The aim of this study is to develop globally optimized generalized moveout approximation in reducing residual normal moveout in VTI media. A comparative analysis was carried out to recent method for given an ellipticity parameter (0 ≤ η ≥ 0.5 ) and wide offset depth to ratio (0 to 4). The result shows that the globally optimized generalized moveout approximation is better in reducing residual moveout at large offsets with a stronger anellipticity parameter than existing methods. This is essential for reducing the accumulation of error especially for deeper substructures.

The conventional normal‐moveout (NMO) and common‐midpoint (CMP) stacking process enhances reflections having a particular moveout velocity, while attenuating events (such as multiple reflections) having different moveout velocities. Unfortunately, this process also acts as a dip filter applied to the CMP stack. In other words, NMO and stacking enhances reflections having a particular slope in the CMP stack, while attenuating reflections having different slopes. NMO and stacking, like any dip filter, degrades lateral resolution. Fortunately, this dip‐filtering action can be suppressed by applying, in addition to NMO, a prestack process known variously as DEVILISH, prestack partial migration, and dip‐moveout. As the latter term implies, this process is a dip‐dependent moveout correction that enables reflections from both horizontal and dipping reflectors to be stacked with the same NMO velocity. Stated another way, NMO velocities estimated from dip‐moveout‐corrected seismograms are independent of the dips of subsurface reflectors. Dip‐moveout by Fourier transform is a method for performing dip‐moveout (DMO) correction in the frequency‐wavenumber domain. The implementation of this method, which resembles the implementation of a discrete Fourier transform, is quite different from and compares favorably with previously published finite‐difference DMO algorithms. DMO by Fourier transform, unlike DMO by finite‐differences, is accurate for all offsets and all dips, provided that velocity is constant. Because velocity is never constant, some accuracy is inevitably lost; but the application of DMO by Fourier transform to recorded seismograms demonstrates the ability of this process to enhance (1) the dip bandwidth of CMP stacks and (2) the accuracy of velocity estimates.

Conventional prestack migration does not preserve local directional information of the seismic waves at the image points. New attempts such as sectored migration of azimuth-limited or common-offset-vector data only concern source-receiver azimuth and offset on the surface, which can be poor representation of subsurface wavepath direction. Moreover, they could result in inaccurate imaging because they do not account for the energy propagation between azimuths or offset-vectors. In the past decade, local angle-domain seismic imaging has been highly advocated to avoid migration artifacts and to improve velocity estimation in complex media. Considering prestack time migration (PSTM) is still widely used in seismic imaging and seismic data preconditioning for amplitude variations with offset or incident-angle (AVO/AVA) analysis, fracture detection, and reservoir characterization, we present an azimuth-preserved local angle-domain Kirchhoff PSTM approach for such purposes. We apply a seismic imaging condition in 3D local angle domain and use extended superposition of impulse responses retaining subsurface angular attributes, which are evaluated through the incident and scattering phase slowness vectors using classical-diffraction moveout equations in isotropic, vertical transversely isotropic (VTI) and azimuthally anisotropic media. Two-dimensional synthetic examples demonstrate what the migrated results look like in local angle domain. A wide-azimuth synthetic example with horizontal transversely isotropy (HTI) proves the necessity of azimuthal migration for reliable imaging and azimuthal analysis when azimuthal anisotropy exists in the overburden. Real data examples show the advantages of imaging in subsurface angle domain for properly focusing and revealing azimuth- and angle-dependent variations of residual moveout and migrated amplitudes.

Progress in seismic inversion and processing in anisotropic media depends on our ability to relate different seismic signatures to the anisotropic parameters. While the conventional notation (stiffness coefficients) is suitable for forward modeling, it is inconvenient in developing analytic insight into the influence of anisotropy on wave propagation. Here, a consistent description of P-wave signatures in transversely isotropic (TI) media with arbitrary strength of the anisotropy is given in terms of Thomsen notation. The influence of transverse isotropy on P-wave propagation is shown to be practically independent of the vertical S-wave velocity VS0, even in models with strong velocity variations. Therefore, the contribution of transverse isotropy to P-wave kinematic and dynamic signatures is controlled by just two anisotropic parameters, ε and δ, with the vertical velocity VP0 being a scaling coefficient in homogeneous models. The distortions of reflection moveouts and amplitudes are not necessarily correlated with the magnitude of velocity anisotropy. The influence of transverse isotropy on P-wave normal-moveout (NMO) velocity in a horizontally layered medium, on small-angle reflection coefficient, and on point-force radiation in the symmetry direction is entirely determined by the parameter δ. Another group of signatures of interest in reflection seisimology–the dip-dependence of NMO velocity, magnitude of nonhyperbolic moveout, time-migration impulse response, and the radiation pattern near vertical–is dependent on both anisotropic parameters (ε and δ) and is primarily governed by the difference between ε and δ. Since P-wave signatures are so sensitive to the value of ε − δ, application of the elliptical-anisotropy approximation (ε = δ) in P-wave processing may lead to significant errors. Many analytic expressions given in the paper remain valid in transversely isotropic media with a tilted symmetry axis. Moreover, the equation for NMO velocity from dipping reflectors, as well as the nonhyperbolic moveout equation, can be used in symmetry planes of any anisotropic media (e.g., orthorhombic).

The elastic constants that control Pand SV-wave propagation in a transversely isotropic media can be estimated by using Pand SV-wave traveltimes from either cross-well or VSP geometries. The procedure consists of two steps. First, elliptical velocity models are used to fit the traveltimes near one axis. The result is four elliptical parameters that represent direct and normal moveout velocities near the chosen axis for Pand SV-waves. Second, the elliptical parameters are used to solve a system of four equations and four unknown elastic constants. The system of equations is solved analytically yielding simple expressions for the elastic constants as a function of direct and normal moveout velocities. For S&waves the estimation of the corresponding elastic constants is easier because the phase velocity is already elliptical. The procedure, introduced for homogeneous media, is generalized to heterogeneous media by using tomographic techniques. The technique is illustrated with synthetic and field data examples from cross-well geometries.

We consider a case where full-azimuth reflection angle gathers were generated using a background VTI depth model. Residual moveouts (RMO) which were automatically picked on these 3D gathers along major horizons indicate considerable periodic azimuthal variations. Our aim is to use the azimuthally dependent RMOs to convert the background VTI model into interval orthorhombic layer parameters. Our method is based on a newly derived generalized Dix-based theory, assuming a locally varying 1D orthorhombic model, where at each location the vertical axis is the same for all layers but the azimuthal orientations are different. An effective model for such a layered structure represents a single layer with identical vertical time, effective compression vertical velocity, effective Thomsen parameters and an effective orientation. The NMO velocity and the surface offset azimuth of the effective model coincide with the parameters of the original package of layers for any azimuth of the phase velocity. Our approach starts with a Fourier-based conversion of the RMOs into azimuthally dependent NMO velocities, which are then inverted into three effective parameters. Finally, we apply the proposed generalized Dix-based inversion approach to estimate the interval orthorhombic parameters within each layer.