Generalized Moveout Approximation Research Articles

The accuracy of approximating P-wave traveltimes in orthotropic media using VTI equations

SUMMARY Analysis of azimuthally varying non-hyperbolic moveout of P-wave traveltimes in orthotropic media can provide valuable information for reservoir fracture characterization and seismic data processing. Moveout velocities and azimuthal orientation of the two vertical symmetry planes plus anellipticity parameters defined for the three symmetry planes of an orthotropic medium can be estimated from the inversion of P-wave traveltimes measured in long-spread, wide-azimuth seismic surveys. Several orthotropic moveout equations have been developed to this date that can be used in the traveltime data inversion. The accuracy of the inversion results is primarily determined by the complexity of the equation and the assumptions (i.e. elastic or acoustic) made to derive it. Here, we show that often simpler moveout equations derived for transverse isotropy media with vertical axis of symmetry (VTI) can be used to estimate anellipticity parameters of the orthotropic media with high accuracy. We make an accuracy comparison analysis between non-hyperbolic moveout equations that are widely used in modelling P-wave traveltimes in VTI media. Our comparison analysis is based on the accuracy of estimating anellipticity parameters defined for two vertical [η(1) and η(2)] and one horizontal [η(3)] symmetry planes of a wide range of orthotropic media. The anellipticity parameters are estimated from the inversion of numerical traveltimes that are generated for a single and horizontal orthotropic layer. We show that for the survey geometry used in this analysis, most of the moveout equations can produce reliable estimates of the anellipticity parameters η(1) and η(2). Despite the availability of traveltime data from wide-azimuths and long source–receiver offsets, none of the moveout equations can estimate η(3) as accurate as η(1) and η(2) estimations. We show that while all the equations produce mostly accurate estimates of η(1) and η(2), generalized moveout approximation (GMA) provides the greatest accuracy. We also assess the validity of using GMA to invert P-wave traveltimes for three anellipticity parameters in terms of anisotropy in the background medium, fracture weakness, source–receiver offset (group angle) and traveltime data coverage in both the vertical (dip) and horizontal (azimuthal) planes. We show that, for a reasonable offset range, GMA can be reliably used to estimate anellipticity parameters η(1) and η(2) for an orthotropic medium that is constructed by intensive fracturing of a host medium with strong VTI anisotropy.

Perturbation-based moveout approximation in the elastic transversely isotropic media with a vertical symmetry axis

Explicit approximations in traveltime are usually used in seismic data processing such as simulation through ray tracing, velocity analysis for model building, and imaging from time migration. Accurate approximation is key to achieving high-resolution imaging of the structure. We have proposed a perturbation-based approximation for P- and S-waves traveltime in a transversely isotropic medium with a vertical symmetry axis. The perturbation coefficients in this approximation are derived directly from the implicit parametric offset-traveltime equation. The comparison with the generalized moveout approximation (GMA) for P and S waves is applied in numerical examples. We find that our proposed approximation indicates higher accuracy than the GMA counterpart for P and S waves. The error sensitivity of P and S waves in the depth, vertical velocity ratio, and the anellipticity parameter also is analyzed to examine the influence of these factors. In addition, we investigate the anisotropy estimation through two approximations for P and S waves and find that our proposed perturbation method results in high accuracy, which can significantly improve the data processing quality. For the S wave, our approximation also is more accurate than the GMA counterpart and could be implemented for joint tomographic inversions.

NMO stretch in vertical transverse isotropic media presented by nonhyperbolic approximations

Normal-moveout (NMO) correction is an inevitable part of routine seismic data processing and imaging. NMO correction suffers from an undesirable phenomenon called NMO stretching, which is more pronounced in larger emerging angles. Frequency distortions caused by NMO stretching for isotropic media have fully been studied in the literature. We express the distortions for the simplest case of anisotropy, which is vertical transverse isotropic media. This is done through different nonhyperbolic approximations, including shifted-hyperbola approximation, rational approximation, and generalized moveout approximation (GMA). To evaluate the predicted distortions, we define a trace-muting threshold and use it in the trace-muting procedure for synthetic data, as well as a field recorded example. Comparison between the results proves that the GMA has the highest accuracy among the other presented approximations. The proposed approach has an immediate potential to be used in industry and also to help to develop innovative nonstretch NMO correction workflows.

Improved generalized moveout approximation with a novel parameterization

Reflection moveout approximations written as explicit functions of offset are convenient tools commonly applied in seismic data modeling, processing, and inversion. The generalized moveout approximation (GMA) has a flexible functional form that accommodates various causes of moveout nonhyperbolicity. We have developed a novel parameterization for the existing six-parameter version of GMA by changing the definition of one of its parameters. Three parameters of the proposed GMA approximation are obtained by fitting the traveltime, the normal moveout velocity, and the fourth-order Taylor series coefficient of traveltime squared at zero offset. The remaining three parameters are obtained by fitting kinematic properties at another two nonzero reference offsets. Through numerical tests on linear sloth models, circular reflector models, and acoustic transversely isotropic media with a vertical symmetry axis (acoustic VTI media), the proposed new GMA method is compared with other published GMA methods in terms of their accuracy. Numerical tests on the three models demonstrate that, when the far reference offset is fixed, the location of the middle reference offset has a considerable impact on the proposed GMA accuracy. We propose a moveout inversion method in a layer-stripping fashion for layered acoustic VTI media based on the proposed and other published GMA methods.

Nonhyperbolic normal moveout stretch correction with deep learning automation

Normal-moveout (NMO) correction is a fundamental step in seismic data processing. It consists of mapping seismic data from recorded traveltimes to corresponding zero-offset times. This process produces wavelet stretching as an undesired by-product. We have addressed the NMO stretching problem with two methods: (1) an exact stretch-free NMO correction that prevents the stretching of primary reflections and (2) an approximate post-NMO stretch correction. Our stretch-free NMO produces parallel moveout trajectories for primary reflections. Our post-NMO stretch correction calculates the moveout of stretched wavelets as a function of offset. Both methods are based on the generalized moveout approximation and are suitable for application in complex anisotropic or heterogeneous environments. We use new moveout equations, modify the original parameter functions to be a constant over the primary reflections, and then interpolate the seismogram amplitudes at the calculated traveltimes. For fast and automatic modification of the parameter functions, we use deep learning. We design a deep neural network (DNN) using convolutional layers and residual blocks. To train the DNN, we generate a set of 40,000 synthetic NMO-corrected common-midpoint gathers and the corresponding desired outputs of the DNN. The data set is generated using different velocity profiles, wavelets, and offset vectors, and it includes multiples, ground roll, and band-limited random noise. The simplicity of the DNN task — a 1D identification of primary reflections — improves the generalization in practice. We use the trained DNN and find successful applications of our stretch-correction method on synthetic and different real data sets.

Approximations for traveltime, slope, curvature, and geometric spreading of elastic waves in layered transversely isotropic media

Each seismic body wave, including quasi-, P-, S-, and converted wave modes, carries useful subsurface information. For the processing, imaging, amplitude analysis, and forward modeling of each wave mode, we need approximate equations of the traveltime, slope (ray parameter), and curvature as a function of offset. Considering the large offset coverage of modern seismic acquisitions, we have developed new approximations designed to be accurate at zero and infinitely large offsets over layered transversely isotropic media with vertical symmetry axis. Our approximation for traveltime is a modified version of the extended generalized moveout approximation that comprises six parameters. Our direct approximations for the ray parameter and curvature use new algebraically simple equations with three parameters. We define these parameters for each wave mode without ray tracing so that we have similar approximate equations for all wave modes that only change based on the parameter definitions. However, our approximations are unable to reproduce S-wave triplications that may occur in some strongly anisotropic models. Using our direct approximation of traveltime derivatives, we also obtain a new expression for the relative geometric spreading. We determine the high accuracy of our approximations using numerical tests on a set of randomly generated multilayer models. Using synthetic data, we show simple applications of our approximations for the normal moveout correction and relative geometric spreading compensation of different wave modes.

Reducing Residual Moveout Seismic Anisotropy Model Using Three-Ray GMA (General Moveout Approximation)

A “ hockey stick” phenomenon is one of anisotropic effects that should be eliminated in marine seismic data. It can increase residual moveout at the far offsets and impact to the distortion of refl ection event amplitude, eventually, reduce the seismic imaging quality. Conventional hyperbolic moveout approximation, an algorithm isotropic model commonly used for seismic processing, has a drawback in supressing such phenomenon. It is also not reliable for medium anisotropy model and long offset data. Many researchers formulated nonhyperbolic moveout approimations but it has limitation analysis for inteval offset-depth ratio (ODR) more than four. We present three-ray generalized moveout approximation (three-ray GMA) for transversely isotropic medium with vertical axis of symmetry (VTI), which is a modifi ed non-hyperbolic moveout approximation from original GMA, to cover up of the weakness of the hyperbolic approximation. The objective of this study is to eliminate “ hockey stick” effect and minimize the residual moveout much smaller at once at the far offsets (offsetdepth ratio 4). In this study, we used synthetic data for single layer model in VTI medium to calculate relative traveltime error for each recent method over a range of offsets (0 ≤ ODR ≤ 6) and anisotropic parameters (0 ≤  ≤ 0.5). We also make comparative method for multi layer and implement it in a velocity analysis and residual moveout calculation. The three-ray GMA shows a better capability than comparative method to reduce residual moveout for larger offset. This result is important for enhancing seismic imaging.

Reducing residual normal moveout by globally optimized generalized moveout approximation in vertically transverse isotropy (VTI) media

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.

Probabilistic moveout analysis by time warping

Parameter estimation from reflection moveout analysis represents one of the most fundamental problems in subsurface model building. We have developed an efficient moveout inversion method based on the process of automatic flattening of common-midpoint (CMP) gathers using local slopes. We find that as a by-product of this flattening process, we can also estimate reflection traveltimes corresponding to the flattened CMP gathers. This traveltime information allows us to construct a highly overdetermined system and subsequently invert for moveout parameters including normal-moveout velocities and quartic coefficients related to anisotropy. We use the 3D generalized moveout approximation (GMA), which can accurately capture the effects of complex anisotropy on reflection traveltimes as the basis for our moveout inversion. Due to the cheap forward traveltime computations by GMA, we use a Monte Carlo inversion scheme for improved handling of the nonlinearity between the reflection traveltimes and moveout parameters. This choice also allows us to set up a probabilistic inversion workflow within a Bayesian framework, in which we can obtain the posterior probability distributions that contain valuable statistical information on estimated parameters such as uncertainty and correlations. We use synthetic and real data examples including the data from the SEAM Phase II unconventional reservoir model to demonstrate the performance of our method and discuss insights into the problem of moveout inversion gained from analyzing the posterior probability distributions. Our results suggest that the solutions to the problem of traveltime-only moveout inversion from 2D CMP gathers are relatively well constrained by the data. However, parameter estimation from 3D CMP gathers associated with more moveout parameters and complex anisotropic models are generally nonunique, and there are trade-offs among inverted parameters, especially the quartic coefficients.

Three-parameter normal moveout correction in layered anisotropic media: A stretch-free approach

In conventional normal moveout (NMO) correction, some parts of the recorded data at larger offsets are discarded because of NMO distortions. Deviation from the true traveltime of reflections due to the anisotropy and heterogeneity of the earth, and wavelet stretching are two reasons of these distortions. The magnitudes of both problems increase with increasing the offset to depth ratio. Therefore, to be able to keep larger offsets of shallower reflections, both problems should be obviated. Accordingly, first, we have studied different traveltime approximations being in use, alongside new parameterizations for two classical functional equations, to select suitable equations for NMO correction. We numerically quantify the fitting accuracy and uncertainty of known nonhyperbolic traveltime approximations for P-waves in transversely isotropic media with vertical symmetry axis (VTI). We select three suitable three-parameter approximations for NMO in layered VTI media as the VTI generalized moveout approximation, a double-square-root approximation, and a perturbation-based approximation. Second, we have developed an extension of the earlier proposed stretch-free NMO method, using the selected moveout approximations. This method involves an automatic modification of the input parameters in anisotropic NMO correction, for selected reflections. Our anisotropic stretch-free NMO method is tested on synthetic and three real data sets from Gulf of Mexico and Iranian oil fields. The results verify the success of the method in extending the usable offsets, by generating flat and stretch-free NMO corrected reflections.

Slowness‐domain kinematical characteristics for horizontally layered orthorhombic media. Part I: Critical slowness match

ABSTRACTKinematical characteristics of reflected waves in anisotropic elastic media play an important role in the seismic imaging workflow. Considering compressional and converted waves, we derive new, azimuthally dependent, slowness‐domain approximations for the kinematical characteristics of reflected waves (radial and transverse offsets, intercept time and traveltime) for layered orthorhombic media with varying azimuth of the vertical symmetry planes. The proposed method can be considered an extension of the well‐known ‘generalized moveout approximation’ in the slowness domain, from azimuthally isotropic to azimuthally anisotropic models. For each slowness azimuth, the approximations hold for a wide angle range, combining power series coefficients in the vicinity of both the normal‐incidence ray and an additional wide‐angle ray. We consider two cases for the wide‐angle ray: a ‘critical slowness match’ and a ‘pre‐critical slowness match’ studied in Parts I and II of this work, respectively. For the critical slowness match, the approximations are valid within the entire slowness range, up to the critical slowness. For the ‘pre‐critical slowness match’, the approximations are valid only within the bounded slowness range; however, the accuracy within the defined range is higher. The critical slowness match is particularly effective when the subsurface model includes a dominant high‐velocity layer where, for nearly critical slowness values, the propagation in this layer is almost horizontal. Comparing the approximated kinematical characteristics with those computed by numerical ray tracing, we demonstrate high accuracy.

A new parameterization for generalized moveout approximation, based on three rays

ABSTRACTA simple and accurate traveltime approximation is important in many applications in seismic data processing, inversion and modelling stages. Generalized moveout approximation is an explicit equation that approximates reflection traveltimes in general two‐dimensional models. Definition of its five parameters can be done from properties of finite offset rays, for general models, or by explicit calculation from model properties, for specific models. Two versions of classical finite‐offset parameterization for this approximation use traveltime and traveltime derivatives of two rays to define five parameters, which makes them asymmetrical. Using a third ray, we propose a balance between the number of rays and the order of traveltime derivatives. Our tests using different models also show the higher accuracy of the proposed method. For acoustic transversely isotropic media with a vertical symmetry axis, we calculate a new moveout approximation in the generalized moveout approximation functional form, which is explicitly defined by three independent parameters of zero‐offset two‐way time, normal moveout velocity and anellipticity parameter. Our test shows that the maximum error of the proposed transversely isotropic moveout approximation is about 1/6 to 1/8 of that of the moveout approximation that had been reported as the most accurate approximation in these media. The higher accuracy is the result of a novel parameterization that do not add any computational complexity. We show a simple example of its application on synthetic seismic data.

Generalized velocity approximation

We have developed a new approximation for P-wave velocity that has the same functional form for phase and group velocities. We call it the generalized velocity approximation (GVA) because it is similar to the generalized moveout approximation. The 2D GVA has five parameters, and the 3D GVA has 12 parameters. Our approximation is exact for the phase velocity in a transversely isotropic medium. The parameters of the 3D version of the proposed approximation are defined in all of the symmetry planes in the same fashion. Numerical examples indicate that our approximation has the same accuracy as recently proposed anelliptic approximation, but it has fewer parameters.

Comparisons of moveout approximations and inversions for P- and S-waves in acoustic transversely isotropic media with a vertical symmetry axis

Acoustic transversely isotropic medium with a vertical symmetry axis (acoustic VTI) is a very practical anisotropic model for P-wave kinematic analysis. However, apart from P-wave, there is also one additional wave that propagates in this medium, widely known as S-wave artifact. This S-wave possesses bizarre properties and is thoroughly different from the conventional shear waves in elastic VTI media. This paper is devoted to the analysis of S-wave artifact moveout approximations and the Dix-type inversion in acoustic VTI media. We consider four different traveltime approximations originally developed for P-wave and apply them for S-wave moveout. The accuracies of P- and S-waves approximations are numerically compared for both homogeneous and multi-layered acoustic VTI models. With an additional reference ray, the generalized moveout approximation performs the best for S-wave traveltime for a large range of offsets. From the effective parameters estimated for P- and S-waves moveout approximations, we also estimate the interval layer parameters from corresponding Dix-type equations developed for P- and S-waves, respectively. The accuracy and feasibility of the Dix-type inversions are demonstrated in a multi-layered model test.

3D generalized nonhyperboloidal moveout approximation

Moveout approximations are commonly used in velocity analysis and time-domain seismic imaging. We revisit the previously proposed generalized nonhyperbolic moveout approximation and develop its extension to the 3D multiazimuth case. The advantages of the generalized moveout approximation are its high accuracy and its ability to reduce to several other known approximations with particular choices of parameters. The proposed 3D functional form involves 17 independent parameters instead of five as in the 2D case. These parameters can be defined by zero-offset traveltime attributes and four additional far-offset rays. In our tests, the proposed approximation achieves significantly higher accuracy than previously proposed 3D approximations.

Analytic calculation of phase and group velocities of P-waves in orthorhombic media

We have developed an approximate method to calculate the P-wave phase and group velocities for orthorhombic media. Two forms of analytic approximations for P-wave velocities in orthorhombic media were built by analogy with the five-parameter moveout approximation and the four-parameter velocity approximation for transversely isotropic media, respectively. They are called the generalized moveout approximation (GMA)-type approximation and the Fomel approximation, respectively. We have developed approximations for elastic and acoustic orthorhombic media. We have characterized the elastic orthorhombic media in Voigt notation, and we can describe the acoustic orthorhombic media by introducing the modified Alkhalifah’s notation. Our numerical evaluations indicate that the GMA-type and Fomel approximations are accurate for elastic and acoustic orthorhombic media with strong anisotropy, and the GMA-type approximation is comparable with the approximation recently proposed by Sripanich and Fomel. Potential applications of the proposed approximations include forward modeling and migration based on the dispersion relation and the forward traveltime calculation for seismic tomography.

Generalized moveout approximation for P–SV converted waves in vertically inhomogeneous transversely isotropic media with a vertical symmetry axis

ABSTRACTWe present an overall description of moveout formulas of P–SV converted waves in vertically inhomogeneous transversely isotropic media with a vertical symmetry axis by using the generalized moveout approximation. The term “generalized” means that this approximation can be reduced to some existing approximations by specific selections of parameters, which provides flexibility in application depending on objectives. The generalized moveout approximation is separately expressed in the phase and group domains. All five parameters of the group domain (or phase domain) generalized moveout approximation are determined using the zero offset (or horizontal slowness) ray and an additional nonzero offset (or horizontal slowness) ray. We discuss the selection of parameters to link the generalized moveout approximation to some existing approximations. The approximations presented are tested on homogeneous, factorized, and layered transversely isotropic models. The results illustrate that utilizing an additional reference ray significantly improves the accuracy of phase‐domain and group‐domain moveout approximations for a large range of horizontal slownesses and source–receiver offsets.

Mapping of moveout attributes using local slopes

Decomposing seismic data in local slopes is the basic idea behind velocity-independent imaging. Using accurate moveout approximations enables computing moveout attributes such as normal moveout velocity and nonhyperbolic parameters as functions of zero-offset travel time. Mapping of moveout attributes is performed from the pre-stack seismic data domain into the time-migrated image domain. The different moveout attributes have different accuracy for a given moveout approximation that depends on the corresponding order of travel-time derivative. The most accurate attribute is the zero-offset travel time, and the nonhyperbolic parameter has the worst accuracy, regardless of the moveout approximation. Typically, the mapping of moveout attributes is performed using a point-to-point procedure, whereas the generalized moveout approximation requires two point-to-point mappings. Testing the attribute mapping on the different models shows that the accuracy of mapped attributes is model dependent, whereas the generalized moveout approximation gives practically exact results.

Moveout approximation for vertical seismic profile geometry in a 2D model with anisotropic layers

ABSTRACTI introduce a new explicit form of vertical seismic profile (VSP) traveltime approximation for a 2D model with non‐horizontal boundaries and anisotropic layers. The goal of the new approximation is to dramatically decrease the cost of time calculations by reducing the number of calculated rays in a complex multi‐layered anisotropic model for VSP walkaway data with many sources. This traveltime approximation extends the generalized moveout approximation proposed by Fomel and Stovas. The new equation is designed for borehole seismic geometry where the receivers are placed in a well while the sources are on the surface. For this, the time‐offset function is presented as a sum of odd and even functions. Coefficients in this approximation are determined by calculating the traveltime and its first‐ and second‐order derivatives at five specific rays. Once these coefficients are determined, the traveltimes at other rays are calculated by this approximation. Testing this new approximation on a 2D anisotropic model with dipping boundaries shows its very high accuracy for offsets three times the reflector depths. The new approximation can be used for 2D anisotropic models with tilted symmetry axes for practical VSP geometry calculations. The new explicit approximation eliminates the need of massive ray tracing in a complicated velocity model for multi‐source VSP surveys. This method is designed not for NMO correction but for replacing conventional ray tracing for time calculations.

Generalized moveout approximation for qP- and qSV-waves in a homogeneous transversely isotropic medium

The moveout approximations can be used in kinematic modeling, velocity analysis, and time migration. The generalized moveout approximation involves five approximation parameters and has several known approximations as special cases. A method is demonstrated for determining parameters of the generalized nonhyperbolic moveout approximation for qP- and qSV-waves in a homogeneous transversely isotropic medium with vertical symmetry axis (VTI medium). The additional parameters for the generalized approximation are computed from the hyperbolic asymptote at infinite offset. Comparison with a few well-known moveout approximations for higher-order terms in the Taylor series and asymptotic behavior shows that the generalized moveout approximation is superior to other nonhyperbolic approximations. A few numerical examples for qP- and qSV-waves in a VTI medium also indicate that the generalized approximation performs the best.