Traveltime Approximation Research Articles

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.

Velocity inversion by global optimization using finite-offset common-reflection-surface stacking applied to synthetic and Tacutu Basin seismic data

Estimation of an accurate velocity macromodel is an important step in seismic imaging. We have developed an approach based on coherence measurements and finite-offset (FO) beam stacking. The algorithm is an FO common-reflection-surface tomography, which aims to determine the best layered depth-velocity model by finding the model that maximizes a semblance objective function calculated from the amplitudes in common-midpoint (CMP) gathers stacked over a predetermined aperture. We develop the subsurface velocity model with a stack of layers separated by smooth interfaces. The algorithm is applied layer by layer from the top downward in four steps per layer. First, by automatic or manual picking, we estimate the reflection times of events that describe the interfaces in a time-migrated section. Second, we convert these times to depth using the velocity model via application of Dix’s formula and the image rays to the events. Third, by using ray tracing, we calculate kinematic parameters along the central ray and build a paraxial FO traveltime approximation for the FO common-reflection-surface method. Finally, starting from CMP gathers, we calculate the semblance of the selected events using this paraxial traveltime approximation. After repeating this algorithm for all selected CMP gathers, we use the mean semblance values as an objective function for the target layer. When this coherence measure is maximized, the model is accepted and the process is completed. Otherwise, the process restarts from step two with the updated velocity model. Because the inverse problem we are solving is nonlinear, we use very fast simulated annealing to search the velocity parameters in the target layers. We test the method on synthetic and real data sets to study its use and advantages.

Three-parameter Radon transform in layered transversely isotropic media

Radon transform is a powerful tool with many applications in different stages of seismic data processing, because of its capability to focus seismic events in the transform domain. Three-parameter Radon transform can optimally focus and separate different seismic events, if its basis functions accurately match the events. In anisotropic media, the conventional hyperbolic or shifted hyperbolic basis functions lose their accuracy and cannot preserve data fidelity, especially at large offsets. To address this issue, we propose an accurate traveltime approximation for transversely isotropic media with vertical symmetry axis, and derive two versions of Radon basis functions, timevariant and time-invariant. A time-variant basis function can be used in time domain Radon transform algorithms while a time-invariant version can be used in, generally more efficient, frequency domain algorithms. Comparing the time-variant and time-invariant Radon transform by the proposed basis functions, the time-invariant version can better focus different seismic events; it is also more accurate, especially in presence of vertical heterogeneity. However, the proposed time-invariant basis functions are suitable for a specific type of layered anisotropic media, known as factorized media. We test the proposed methods and illustrate successful applications of them for trace interpolation and coherent noise attenuation.

VSP data inversion for vertical velocity gradient and elliptical anisotropy model

Inversion of velocity parameters for the walkaway VSP data in a multilayered medium can be impeded by velocity gradients and anisotropy in some layers. A problem occurs if we compare velocities obtained from borehole seismic profiling which are equal to their vertical components with the velocities calculated with paths coming from far offsets where the horizontal component plays an important role, especially when the vertical gradient exists and the ray paths are curve-shaped. In this contribution we present the results of velocity model inversion for VSP data considering velocity gradient and elliptical anisotropy. The algorithm consists of two steps, optimization of velocity parameters and optimization of ray paths for the given model. Both procedures use the Nelder-Mead simplex method which finds local minima. Due to the character of optimization we performed also multistart analysis which can provide information about possible equivalences between parameters. Analysis was conducted for different parameterizations, in some cases allowing introduction of additional parameters: vertical gradient and elliptical anisotropy coefficient. The optimal model for a specific set of data is chosen with the help of Bayesian Information Criterion to balance complexity of model with quality of approximation of traveltimes.

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.

RELATION BETWEEN THE MODEL AND THE TOPOGRAPHY OF THE OBJECTIVE FUNCTION IN A VELOCITY ANALYSIS USING A NONHYPERBOLIC MULTICOMPONENT TRAVEL-TIME APPROXIMATION

The velocity analysis is an important step for the seismic processing. With the increase in the difficulty of work with specific conditions and complex geological structures, more complex travel-time approximations are developed to perform a better structural characterization. As the complexity increases, more approximations were developed aiming to characterize different factor responsible for the nonhyperbolicity, and using sometimes an additional parameter. Many nonhyperbolic multiparametric travel-time approximations were developed and their complexities vary strongly from one to other. In previous works, it was observed that some approximations present a unimodal behavior while other approximations showed a multimodal behavior. However, a specific kind of approximation showed both statistical distributions, the unimodal and the multimodal in distinct situations. To find out the factor responsible for this variation in the probabilistic distribution behavior of this kind of approximation, it was necessary to test different theoretical models to understand which variation in the structure interferes in the topography of the objective function.

Traveltime approximation for strongly anisotropic media using the homotopy analysis method

Traveltime approximation for strongly anisotropic media using the homotopy analysis method

Stretch‐free generalized normal moveout correction

ABSTRACTThe effective application of normal moveout correction processes mainly depends on four factors: the chosen traveltime approximation, the stretching associated with the given traveltime, crossing events and phase changes, the last two being inherent to the seismic data. In this context, we conduct a quantitative analysis on stretching considering a general traveltime expression depending on half‐offset and midpoint coordinates. Through this analysis, we propose a mathematically proven procedure to eliminate stretching, which can be applied to any traveltime approximation. The proposed method is applied to synthetic and real data sets, considering different traveltime approximations and achieved complete elimination of stretching.

Slowness domain offset and traveltime approximations in layered vertical transversely isotropic media

ABSTRACTConsidering horizontally layered transversely isotropic media with vertical symmetry axis and all types of pure‐mode and converted waves we present a new wide‐angle series approximation for the kinematical characteristics of reflected waves: horizontal offset, intercept time, and total reflection traveltime as functions of horizontal slowness. The method is based on combining (gluing) both zero‐offset and (large) finite‐offset series coefficients. The horizontal slowness is bounded by the critical value, characterised by nearly horizontal propagation within the layer with the highest horizontal velocity. The suggested approximation uses five parameters to approximate the offset, six parameters to approximate the intercept time or the traveltime, and seven parameters to approximate any two or all three kinematical characteristics. Overall, the method is very accurate for pure‐mode compressional waves and shear waves polarised in the horizontal plane and for converted waves. The application of the method to pure‐mode shear waves polarised in the vertical plane is limited due to cusps and triplications. To demonstrate the high accuracy of the method, we consider a synthetic, multi‐layer model, and we plot the normalised errors with respect to numerical ray tracing.

Three-parameter Radon transform based on shifted hyperbolas

The focusing power of the conventional hyperbolic Radon transform decreases for long-offset seismic data due to the nonhyperbolic behavior of moveout curves at far offsets. Furthermore, conventional Radon transforms are ineffective for processing data sets containing events of different shapes. The shifted hyperbola is a flexible three-parameter (zero-offset traveltime, slowness, and focusing-depth) function, which is capable of generating linear and hyperbolic shapes and improves the accuracy of the seismic traveltime approximation at far offsets. Radon transform based on shifted hyperbolas thus improves the focus of seismic events in the transform domain. We have developed a new method for effective decomposition of seismic data by using such three-parameter Radon transform. A very fast algorithm is constructed for high-resolution calculations of the new Radon transform using the recently proposed generalized Fourier slice theorem (GFST). The GFST establishes an analytic expression between the [Formula: see text] coefficients of the data and the [Formula: see text] coefficients of its Radon transform, with which a very fast switching between the model and data spaces is possible by means of interpolation procedures and fast Fourier transforms. High performance of the new algorithm is demonstrated on synthetic and real data sets for trace interpolation and linear (ground roll) noise attenuation.

Post‐stack diffraction imaging in vertical transverse isotropy media using non‐hyperbolic moveout approximations

ABSTRACTDiffractions carry valuable information about local discontinuities and small‐scale objects in the subsurface. They are still not commonly used in the process of geological interpretation. Many diffraction imaging techniques have been developed and applied for isotropic media, whereas relatively few techniques have been developed for anisotropic media. Ignoring anisotropy can result in low‐resolution images with wrongly positioned or spurious diffractors. In this article, we suggest taking anisotropy into account in two‐dimensional post‐stack domain by considering P‐wave non‐hyperbolic diffraction traveltime approximations for vertical transverse isotropy media, previously developed for reflection seismology. The accuracy of the final images is directly connected to the accuracy of the diffraction traveltime approximations. We quantified the accuracy of six different approximations, including hyperbolic moveout approximation, by the application of a post‐stack diffraction imaging technique on two‐dimensional synthetic data examples.

Fourth-order normal moveout velocity in elastic layered orthorhombic media — Part 2: Offset-azimuth domain

Based on the theory derived in part 1, in which we obtained the azimuthally dependent fourth-order normal-moveout (NMO) velocity functions for layered orthorhombic media in the slowness-azimuth/slowness and the slowness-azimuth/offset domains, in part 2, we extend the theory to the offset-azimuth/slowness and offset-azimuth/offset domains. We reemphasize 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. The same type of models as in part 1 are considered, in which the layers share a common horizontal plane of symmetry, but the azimuths of their vertical symmetry planes are different. The same eight local (single-layer) and global (overburden multilayer) effective parameters are used. In addition, we have developed an alternative set of global effective parameters in which the “anisotropic” effective parameters are normalized, classified into two groups: two “azimuthally isotropic” parameters and six “azimuthally anisotropic” parameters. These parameters have a clearer physical interpretation and they are suitable for inversion purposes because they can be controlled and constrained. Next, we propose a special case, referred to as “weak azimuthal anisotropy,” in which only the azimuthally anisotropic effective parameters are assumed to be weak. The resulting NMO velocity functions are considerably simplified, reduced to the form of the slowness-azimuth/slowness formula. We verify the correctness of our method by applying it to a multilayer orthorhombic medium with strong anisotropy. We introduce our derived, fourth-order slowness-azimuth/offset domain NMO velocity function into the well-known nonhyperbolic asymptotic traveltime approximation, and we compare the approximate traveltimes with exact traveltimes obtained by two-point ray tracing. The comparison shows an accurate match up to moderate offsets. Although the accuracy with the weak azimuthal anisotropic formula is inferior, it can still be considered reasonable for practical use.

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 slowness-azimuth 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.

Traveltime approximation in vertical transversely isotropic layered media

ABSTRACTThe well‐known asymptotic fractional four‐parameter traveltime approximation and the five‐parameter generalised traveltime approximation in stratified multi‐layer transversely isotropic elastic media with a vertical axis of symmetry have been widely used for pure‐mode and converted waves. The first three parameters of these traveltime expansions are zero‐offset traveltime, normal moveout velocity, and quartic coefficient, ensuring high accuracy of traveltimes at short offsets. The additional parameter within the four‐parameter approximation is an effective horizontal velocity accounting for large offsets, which is important to avoid traveltime divergence at large offsets. The two additional parameters in the above‐mentioned five‐parameter approximation ensure higher accuracy up to a given large finite offset with an exact match at this offset. In this paper, we propose two alternative five‐parameter traveltime approximations, which can be considered extensions of the four‐parameter approximation and an alternative to the five‐parameter approximation previously mentioned. The first three short‐offset parameters are the same as before, but the two additional long‐offset parameters are different and have specific physical meaning. One of them describes the propagation in the high‐velocity layer of the overburden (nearly horizontal propagation in the case of very large offsets), and the other characterises the intercept time corresponding to the critical slowness that includes contributions of the lower velocity layers only. Unlike the above‐mentioned approximations, both of the proposed traveltime approximations converge to the theoretical (asymptotic) linear traveltime at the limit case of very large (“infinite”) offsets. Their accuracy for moderate to very large offsets, for quasi‐compressional waves, converted waves, and shear waves polarised in the horizontal plane, is extremely high in cases where the overburden model contains at least one layer with a dominant higher velocity compared with the other layers. We consider the implementation of the proposed traveltime approximations in all classes of problems in which the above‐mentioned approximations are used, such as reflection and diffraction analysis and imaging.

Seismic data enhancement and regularization using finite offset Common Diffraction Surface (CDS) stack

The Common Reflection Surface stack is a robust method for simulating zero-offset and common-offset sections with high accuracy from multi-coverage seismic data. For simulating common-offset sections, the Common-Reflection-Surface stack method uses a hyperbolic traveltime approximation that depends on five kinematic parameters for each selected sample point of the common-offset section to be simulated. The main challenge of this method is to find a computationally efficient data-driven optimization strategy for accurately determining the five kinematic stacking parameters on which each sample of the stacked common-offset section depends. Several authors have applied multi-step strategies to obtain the optimal parameters by combining different pre-stack data configurations. Recently, other authors used one-step data-driven strategies based on a global optimization for estimating simultaneously the five parameters from multi-midpoint and multi-offset gathers. In order to increase the computational efficiency of the global optimization process, we use in this paper a reduced form of the Common-Reflection-Surface traveltime approximation that depends on only four parameters, the so-called Common Diffraction Surface traveltime approximation. By analyzing the convergence of both objective functions and the data enhancement effect after applying the two traveltime approximations to the Marmousi synthetic dataset and a real land dataset, we conclude that the Common-Diffraction-Surface approximation is more efficient within certain aperture limits and preserves at the same time a high image accuracy. The preserved image quality is also observed in a direct comparison after applying both approximations for simulating common-offset sections on noisy pre-stack data.

Application of perturbation theory to a P-wave eikonal equation in orthorhombic media

The P-wave eikonal equation for orthorhombic (ORT) anisotropic media is a highly nonlinear partial differential equation requiring the solution of a sixth-order polynomial to obtain traveltimes, resulting in complex and time-consuming numerical solutions. To alleviate this complexity, we approximate the solution of this equation by applying a multiparametric perturbation approach. We also investigated the sensitivity of traveltime surfaces in ORT media with respect to three anelliptic parameters. As a result, a simple and accurate P-wave traveltime approximation valid for ORT media was derived. Two different possible anelliptic parameterizations were compared. One of the parameterizations includes anelliptic parameters defined at zero offset: [Formula: see text], [Formula: see text], and [Formula: see text]. Another parameterization includes anelliptic parameters defined for all symmetry planes: [Formula: see text], [Formula: see text], and [Formula: see text]. The azimuthal behavior of sensitivity coefficients with different parameterizations was used to analyze the crosstalk between anelliptic parameters.

Traveltime approximations and parameter estimation for orthorhombic media

Building anisotropy models is necessary for seismic modeling and imaging. However, anisotropy estimation is challenging due to the trade-off between inhomogeneity and anisotropy. Luckily, we can estimate the anisotropy parameters if we relate them analytically to traveltimes. Using perturbation theory, we have developed traveltime approximations for orthorhombic media as explicit functions of the anellipticity parameters [Formula: see text], [Formula: see text], and [Formula: see text] in inhomogeneous background media. The parameter [Formula: see text] is related to Tsvankin-Thomsen notation and ensures easier computation of traveltimes in the background model. Specifically, our expansion assumes an inhomogeneous ellipsoidal anisotropic background model, which can be obtained from well information and stacking velocity analysis. We have used the Shanks transform to enhance the accuracy of the formulas. A homogeneous medium simplification of the traveltime expansion provided a nonhyperbolic moveout description of the traveltime that was more accurate than other derived approximations. Moreover, the formulation provides a computationally efficient tool to solve the eikonal equation of an orthorhombic medium, without any constraints on the background model complexity. Although, the expansion is based on the factorized representation of the perturbation parameters, smooth variations of these parameters (represented as effective values) provides reasonable results. Thus, this formulation provides a mechanism to estimate the three effective parameters [Formula: see text], [Formula: see text], and [Formula: see text]. We have derived Dix-type formulas for orthorhombic medium to convert the effective parameters to their interval values.

Differential evolution-based optimization procedure for automatic estimation of the common-reflection surface traveltime parameters

The common-reflection surface (CRS) method is a sophisticated alternative to the traditional common-midpoint stacking because its traveltime approximation allows for the use of more traces than the normal moveout. This in turn requires more parameters for the moveout description, thus increasing the computational burden of the parameter estimation. In the literature, a suboptimal strategy is often used, which decreases the complexity but, as we found in this work, compromises the accuracy of the parameters in some cases. To cope with this problem, in this work, we have devised detailed information for efficient estimation of the CRS parameters using the differential evolution (DE) global optimization algorithm. Because we used data sets with low fold and low signal-to-noise ratio, from which no reliable velocity analysis could be easily performed, we applied this algorithm in a fully automatic global search, i.e., without any velocity guide. The results for a 2D real data set from Brazil indicated that the global strategy yielded good results, both in terms of image quality as in the quality of the parameter volumes, especially the stacking velocity estimates, while keeping the computational costs relatively low. We also developed a convergence and a sensitivity analysis of the DE that shows its computational efficiency and the robustness of the optimization method with respect to the choice of the control parameters of the algorithm.

Fresnel volume and interface Fresnel zone for reflected and transmitted waves from a curved interface in anisotropic media

The Fresnel volume and the interface Fresnel zone (IFZ) concepts play an important role in seismic exploration because the IFZ largely contributes to the formation of the reflection and transmission wavefields at an observation point. We derived analytic expressions based on traveltime approximations to evaluate the IFZ size for converted and nonconverted waves reflected (or transmitted) by a curved interface between two homogeneous general anisotropic media, and more specifically for dip-constrained transversely isotropic homogeneous media. The reflectors are of anticline, syncline, or saddle type, and their principal curvature axes may not lie in the incidence plane. As in an anisotropic medium the isochron in most cases assumes a nonelliptical shape, the size and the shape of the IFZ for reflected waves are predominantly dependent on the curvatures of the isochrons together with the curvatures of the interface. The IFZ shapes also exhibit large variation with interface curvature and incidence angle. In addition, the difference between the Thomsen anisotropy parameters [Formula: see text] and [Formula: see text] is found to control the size of the IFZ for P-P and P-S reflections. The IFZ for anisotropic media with curved interface can be much larger than that for equivalent isotropic media, and more specifically for positive values of [Formula: see text]. The spatial resolution of unmigrated seismic data in anisotropic media would consequently be different from that determined for the same configuration for isotropic models and a planar interface.

Travel-time approximation of acoustic ranging in GPS/Acoustic seafloor geodesy

Previous studies have proposed methods to reduce the effect of sound-speed variation in the water column on the precision of GPS/Acoustic positioning. However, the fit of temporal travel-time variation makes reference to sound-speed profiles without considering depth-dependent variation. Accordingly, the goal of this study is to provide a simple and precise approximation for a given sound-speed profile considering its depth-dependent variation. We propose two synthetic models for sound-speed profiles, one linear and the other bilinear, for better travel-time approximations of acoustic ranging in GPS/Acoustic seafloor geodesy. The linear and bilinear models are tested on three types of sound-speed profiles derived from conductivity–temperature–depth (CTD) casts made at water depths of 300, 1000, and 2000m, respectively. The change in error of the best acoustic travel-time approximation resulting from the change in break depth of the bilinear model is investigated for the three types of sound-speed profiles. Furthermore, the performance of the linear and bilinear models is evaluated using the data collected from a GPS/Acoustic survey. The evaluation results demonstrate that both the linear and bilinear models can effectively reduce the effect of sound-speed variation on the precision of GPS/Acoustic positioning.