Traveltime Approximation Research Articles

An approximation method to determine the travel time in total focusing imaging of carbon fiber-reinforced laminates

Abstract It is critical to accurately determine the propagation time in testing objects for ultrasonic array imaging using the Total Focusing Method (TFM). However, it poses challenges for the calculation of wave propagation time in carbon fiber reinforced polymer (CFRP) laminates because of the material anisotropy and inhomogeneity. This paper introduces a novel approach for computing ultrasonic wave travel time to enhance the TFM image of CFRP laminates. This method, termed the layer-by-layer acoustic travel time approximation (L-L TravelTimeAppro), considers the anisotropy within each monolayer and the heterogeneity of CFRP. It approximates the propagation path as straight throughout the material and computes the wave propagation duration utilizing the group velocity in each ply. The application of this method to TFM imaging is referred to as TravelTimeAppro-TFM. Experiments were conducted in a specially designed CFRP laminate. Subsequently, TFM imaging was processed by using propagation time calculated by isotropic, Dijkstra, and L-L TravelTimeAppro methods, respectively. The results indicate that TravelTimeAppro-TFM outperforms isotropic-TFM in terms of imaging amplitude gain with a maximum gain of 4.67 dB. Furthermore, it offers superior computational efficiency compared to Dijkstra-TFM.

Dynamic OHT Routing Using Travel Time Approximation Based on Deep Neural Network

Dynamic OHT Routing Using Travel Time Approximation Based on Deep Neural Network

Layer-by-layer acoustic travel time approximation using ray theory for total focusing method imaging in carbon fiber reinforced polymer

This paper proposes a layer-by-layer acoustic travel time approximation method based on ray theory for total focusing method (TFM) imaging in carbon fiber reinforced polymer (CFRP) laminates. The method considers the anisotropy in every monolayer and heterogeneity of CFRP, which approximates the path of propagation as straight in the whole material. The application of this method to TFM imaging is called TravelTimeAppro-TFM. In comparison to isotropic-TFM and Dijkstra-TFM, the experimental results indicated that TravelTimeAppro-TFM outperforms isotropic-TFM in terms of imaging amplitude gain with a maximum gain of 4.67 dB. On the other hand, this approach reduces the computational work compared to Dijkstra-TFM. The proposed method demonstrates significant improvements in both focusing performance and the speed of calculation. This paper also investigates the effective angular range of the layer-by-layer acoustic travel time approximation method through experimental and finite element simulation analysis.

Least-Squares Kirchhoff Depth Migration With Fast Point-Spread-Function Computation

Seismic migrations are generally formulated as the adjoint operators of linear forward modeling and often lead to images with degraded resolution, unbalanced illumination and migration artifacts, especially in surveys with geologic complexity and irregular acquisition geometry. Least-squares migration (LSM) is able to mitigate these problems and produce better resolved images that are suitable for subsequent AVO/AVA inversion. However, no matter what domain LSM is implemented in, the computational cost is still several times or even one order of magnitude more than that of traditional migration. In this paper, we present an efficient image-domain least-square Kirchhoff depth migration (LSKDM), in which the Hessian matrix is approximated by a grid of point-spread-functions (PSFs). Traditional PSF computing algorithm requires a non-negligible cost caused by a successive operation of modeling and migration, and has to satisfy a sampling restriction to avoid interference between nearby PSFs. We present in this paper that, by using the ray-based Green’s functions and the linear traveltime approximation, the PSFs can be constructed explicitly at a significantly reduced computational cost and is able to adapt flexible spatial sampling that is fine enough to detect small-scale illumination variation. With the constructed PSFs, we formulate an image-domain LSKDM to iteratively solve for the optimal reflectivities. Numerical tests on synthetic and field data examples demonstrate that the proposed LSKDM is highly efficient and is capable of producing images with enhanced spatial resolution and amplitude fidelity when compared with the Kirchhoff depth migration (KDM) image.

Comparison between L2- and L1-norm and among optimization algorithms for a nonhyperbolic multiparametric approach for converted-wave and OBN data

As velocity analysis is an important step in seismic processing, several nonhyperbolic travel-time approximations have been proposed during the last decades, and each nonhyperbolic approximation was developed for different conditions and with different proposals. However, none of them was proposed to consider the combined effect of the nonhyperbolicity coming from layered media with large offsets, wave conversion and difference of datum between source and receiver. For this, a nonhyperbolic multiparametric travel-time approximation, which is capable of describing this combination of effects, was recently proposed. As this approximation was developed to characterize ultra-deep reservoirs, the understanding of its behavior is necessary for an offshore reservoir concerning the objective function topology complexity, as it is important for a better understanding of its behavior during the inversion procedure, and also important to determine the kind of optimization algorithm to be used. Performing the inversion procedure with different optimization algorithms and norms is proposed. The complexity analysis of the objective function is also proposed. Then, a comparison between each norm and among each algorithm concerning their accuracy and efficiency is proposed, to find which combination is the most effective to recover RMS velocity information for this kind of scenario.

P-, SV- and SH-wave eikonal approximations for HTI media based on perturbation theory

SUMMARY The eikonal equation for transversely isotropic media is a complex quartic partial differential equation that causes instability in traveltime inversion. The phenomenon of shear wave splitting in a transversely isotropic model with a horizontal symmetry axis (HTI) provides information for inversion. However, the involvement of shear waves makes the computation of forward modelling and inversion more time-consuming than when only using Pwaves. Most studies of traveltime approximation based on the eikonal equation are based on an acoustic assumption, which does not work for shear waves. Therefore, based on perturbation theory, we derive approximate solutions of the eikonal equation in a heterogeneous elastic HTI model for P, SV and SH waves to improve the traveltime computation efficiency. The approximate solutions of linear, quadratic and Shanks forms are based on the linearization of the traveltime function in terms of fracture weaknesses in an isotropic background, which are suitable for joint P-, SV- and SH-wave traveltime inversion and tomography for fracture parameters. When applied for a homogenous model, the proposed formulation results in a simple and accurate traveltime approximation. The first- and second-order perturbation coefficients are used to analyse the sensitivity of the traveltime to fracture weakness. To improve the accuracy of our approximation, we also expand the approximation of the P-wave traveltime with respect to the anellipticity parameter η in an elliptic background, which is evaluated by performing an error test. Based on the accuracy and stability of the perturbation, we select a set of linear equations with respect to the fracture weaknesses and a set of more accurate nonlinear equations to serve different needs.

Perturbation‐based traveltime approximation for two acoustic assumptions in the transversely isotropic media with a vertical symmetry axis

ABSTRACTThe accuracy of an explicit traveltime‐offset approximation affects the results of the velocity analysis that plays a crucial role in the seismic data processing. Seismic anisotropy is important to account for large offset and azimuth since it can provide detailed information comparing with the isotropic assumption. For the perturbation‐based method, different traveltime approximation forms result in different accuracy. It is necessary to find the optimal traveltime approximation for specific signs and magnitudes of the anellipticity and different offset ranges. In this paper, a series of perturbation‐based traveltime approximations from different acoustic assumptions and parameterizations are specified. The perturbation coefficients are derived from the corresponding acoustic eikonal equations. We test the accuracy of the defined perturbation approximations in four homogeneous transversely isotropic medium with vertical symmetry axis (VTI) medium with a different set of anisotropy parameters and one multilayered transversely isotropic medium with vertical symmetry axis medium. The sensitivity in anellipticity for different approximations is also analysed. We find that different approximation achieves different accuracy under the circumstance of specific offset range and the value of anellipticity. Therefore, the optimal approximation form with higher accuracy can be selected based on different offset ranges and the magnitudes of anellipticity.

Large-offset P-wave traveltime in layered transversely isotropic media

Large-offset seismic data processing, imaging, and velocity estimation require an accurate traveltime approximation over a wide range of offsets. In layered transversely isotropic media with a vertical symmetry axis, the accuracy of traditional traveltime approximations is limited to near offsets. We have developed a new traveltime approximation that maintains the accuracy of classic equations around the zero offset and exhibits the correct curvilinear asymptote at infinitely large offsets. Our approximation is based on the conventional acoustic assumption. Its equation incorporates six parameters. To define them, we use the Taylor series expansion of the exact traveltime around the zero offset and a new asymptotic series for the infinite offset. Our asymptotic equation indicates that the traveltime behavior at infinitely large offsets is dominated by the properties of the layer with the maximum horizontal velocity in the sequence. The parameters of our approximation depend on the effective zero-offset traveltime, the normal moveout velocity, the anellipticity, a new large-offset heterogeneity parameter, and the properties of the layer with the maximum horizontal velocity in the sequence. We have applied our traveltime approximation (1) to directly calculate traveltime and ray parameter at given offsets, as analytical forward modeling, and (2) to estimate the first four of the aforementioned parameters for the layers beneath a known high-velocity layer. Our large-offset heterogeneity parameter includes the layering effect on the reflections’ traveltime.

Wave-equation time migration

Time migration, as opposed to depth migration, suffers from two well-known shortcomings: (1) approximate equations are used for computing Green’s functions inside the imaging operator and (2) in case of lateral velocity variations, the transformation between the image-ray coordinates and the Cartesian coordinates is undefined in places where the image rays cross. We have found that the first limitation can be removed entirely by formulating time migration through wave propagation in image-ray coordinates. Our approach constructs a time-migrated image without relying on any kind of traveltime approximation by formulating an appropriate geometrically accurate acoustic wave equation in the time-migration domain. The advantage of this approach is that the propagation velocity in image-ray coordinates does not require expensive model building and can be approximated by quantities that are estimated in conventional time-domain processing. Synthetic and field data examples demonstrate the effectiveness of the proposed approach and show that the proposed imaging workflow leads to a significant uplift in terms of image quality and can bridge the gap between time and depth migrations. The image obtained by the proposed algorithm is correctly focused and mapped to depth coordinates; it is comparable to the image obtained by depth migration.

Rational approximation of P-wave kinematics — Part 2: Orhorhombic media

Orthorhombic anisotropy is a modern standard for 3D seismic studies in complex geologic settings. Several seismic data processing methods and wave propagation modeling algorithms in orthorhombic media rely on phase-velocity, group-velocity, and traveltime approximations. The algebraic simplicity of an approximate equation is an important factor in these media because the governing equations are more complicated than transversely isotropic media. To approximate the P-wave kinematics in acoustic orthorhombic media, we have developed a new 3D general functional equation that has a simple rational form. Using the general form, we adopt two versions of rational approximations for the phase velocity, group velocity, and traveltime. The first version uses a simpler functional form and parameter definition within the orthorhombic symmetry planes. The second version is more accurate, using one parameter that is defined out of the symmetry planes. For the phase velocity, we obtain another approximation that is no longer rational but is still algebraically simple, exact for 3D transversely isotropic media, and it is exact within the symmetry planes of orthorhombic media. We find superior accuracy in our approximations compared with previous ones, using numerical studies on multiple moderately anisotropic orthorhombic models. We investigate the effect of the negative anellipticity parameters on the accuracy and find that, in models in which the error of the existing most accurate approximations exceeds 2%, the error of the new approximations remains below 0.2%. The adopted approximations are algebraically simpler and stably more accurate than existing approximations; therefore, they may be considered as attractive alternatives for the existing approximations in many practical applications. We extend the applicability of our approximations by using them to obtain the equations of group direction as a function of phase direction and vice versa, which are useful in wave propagation modeling methods.

Traveltime-table compression using artificial neural networks for Kirchhoff-migration processing of microseismic data

Massive computation of seismic traveltimes is widely used in seismic processing, for example, for the Kirchhoff migration of seismic and microseismic data. Implementation of the Kirchhoff migration operators uses large precomputed traveltime tables (for all sources, receivers, and densely sampled imaging points). We have tested the idea of using artificial neural networks for approximating these traveltime tables. The neural network has to be trained for each velocity model, but then the whole traveltime table can be compressed by several orders of magnitude (up to six orders) to the size of less than 1 MB. This makes it convenient to store, share, and use such approximations for processing large data volumes. We evaluate some aspects of choosing neural-network architecture, training procedure, and optimal hyperparameters. On synthetic tests, we find a reasonably accurate approximation of traveltimes by neural networks for various velocity models. A final synthetic test shows that using the neural-network traveltime approximation results in good accuracy of microseismic event localization (within the grid step) in the 3D case.

Improving the accuracy of the expanded anisotropic eikonal equation at larger offsets using Levin T-transformation

In an anisotropic model, traveltime can be determined approximately by numerical solution of the eikonal equation in terms of an anellipticity parameter η, using perturbation theory. However, its accuracy decreases under the effect of strong anisotropy at larger offsets. It becomes invalid for determining normal moveout velocity and anellipticity parameter in seismic processing. We propose a new approach using Levin T-transformation to transform the expanded traveltime in the transversely isotropic medium with vertical axis of symmetry (VTI) into rational form. The objective of this study is to provide a new traveltime approximation that is more accurate at larger offsets. In this study, we derive Levin algorithm and determine the optimal value of Levin parameters, which is a key step in achieving better accuracy. In a numerical experiment, we compare the accuracy between Levin T-transformation and second sequence of Shanks transformation in a homogeneous VTI medium. We also implement both approximations in a velocity analysis and stacking traces using synthetic common midpoint gathers on a multilayer earth model. The proposed method shows a superiority in accuracy to existing methods over a range of offsets with offset-to-depth ratio up to 6 and anellipticity parameter 0–0.5.

Separation of PP‐ and PS‐wave reflected seismic data using two‐dimensional finite offset common‐reflection‐surface traveltime approximation

ABSTRACTRecently, the interest in PS‐converted waves has increased for several applications, such as sub‐basalt layer imaging, impedance estimates and amplitude‐versus‐offset analysis. In this study, we consider the problem of separation of PP‐ and PS‐waves from pre‐stacked multicomponent seismic data in two‐dimensional isotropic medium. We aim to demonstrate that the finite‐offset common‐reflection‐surface traveltime approximation is a good alternative for separating PP‐ and PS‐converted waves in common‐offset and common shot configurations by considering a two‐dimensional isotropic medium. The five parameters of the finite‐offset common‐reflection‐surface are firstly estimated through the inversion methodology called very fast simulated annealing, which estimates all parameters simultaneously. Next, the emergence angle, one of the inverted parameters, is used to build an analytical separation function of PP and PS reflection separation based on the wave polarization equations. Once the PP‐ and PS‐converted waves were separated, the sections are stacked to increase the signal‐to‐noise ratio using the special curves derived from finite‐offset common‐reflection‐surface approximation. We applied this methodology to a synthetic dataset from simple‐layered to complex‐structured media. The numerical results showed that the inverted parameters of the finite offset common‐reflection‐surface and the separation function yield good results for separating PP‐ and PS‐converted waves in noisy common‐offset and common shot gathers.

Comparative analysis of nonhyperbolic multiparametric travel-time approximations of multicomponent seismic data considering difference of depth between source and receiver using OBN technology

AbstractThe processing of multicomponent seismic data is already a challenge concerning the velocity analysis. When it is performed for offshore survey, the difficulty increases a lot more with the use of OBN (Ocean Bottom Nodes) technology. The ray tracing asymmetry generated by the wave conversion and the difference of datum between source and receptor are not the only factors which contribute for a strongly nonhyperbolic travel-time event. The layered subsurface models and the large offsets employed in the offshore surveys make the nonhyperbolicity even stronger. Aiming to solve this problem, eight approximations to perform the velocity analysis were tested for two models. The complexity analysis of each nonhyperbolic multiparametric approximation was also studied to understand their behaviors during the optimization process. The relative error between the observed curve and the calculated curve with each approximation was computed for PP and PS reflection events of two models. With these informations, it was possible to determine which approximation is the most reliable one for this kind of models.Keywords: multicomponent, OBN, nonhyperbolic, multiparametric. ResumoO processamento de dados sísmicos multicomponentes já é um desafio com relação à análise de velocidades. Quando realizado para levantamentos marítimos, a dificuldade aumenta muito mais com o uso da tecnologia OBN (Ocean Bottom Nodes). A assimetria no traçado de raios gerada pela conversão de onda e pela diferença de profundidade entre fonte e receptor não são os únicos fatores que contribuem para um evento de tempos de trânsito fortemente não-hiperbólico. Os modelos estratificados de subsuperfície e os grandes afastamentos aplicados nos levantamentos marítimos tornam a não-hiperbolicidade ainda mais forte. Visando resolver este problema, oito aproximações para realizar a análise de velocidades foram testadas para dois modelos. A análise de complexidade de cada aproximação não-hiperbólica multiparamétrica também foi estudada para entender seus comportamentos durante o processo de otimização. Os erros relativos entre as curvas observadas e calculadas com cada aproximação foram calculados para os eventos de reflexão PP e PS dos dois modelos. Com estas informações, foi possível determinar qual aproximação é a mais confiável para estes tipos de modelos.Palavras-chave: multicomponente, OBN; não-hiperbólico, multiparamétrico.

Application of very fast simulated annealing and differential evolution in the search for FO-CRS wavefield attributes

The finite-offset common-reflection-surface (FO-CRS) stack method can be used to simulate any common-offset (CO) seismic section by stacking prestack seismic data along the surfaces defined by the paraxial hyperbolic traveltime approximation. In two dimensions, the FO-CRS stacking operator depends on five kinematic wavefield attributes for every time sample of the target CO section. The main problem with this method is identifying a computationally efficient data-driven search strategy for accurately determining the best set of FO-CRS attributes that produce the optimal coherence measure of the seismic signal in the prestack data. Identifying a global optimization algorithm with the best performance is a challenge when solving this optimization problem. This is because the objective function is multimodal and involves a large volume of data, which leads to high computational costs. We introduced a comparative and competitive study through the application of two global optimization algorithms that simultaneously search the FO-CRS attributes from the prestack seismic data, very fast simulated annealing (VFSA) and the differential evolution (DE). By applying this FO-CRS stack to the Marmousi synthetic seismic data set, we have compared the performances of the two optimization algorithms with regard to their efficiency and effectiveness in estimating the five FO-CRS attributes. To analyze the robustness of the two algorithms, we apply them to real land seismic data and show their ability to find the near-optimal attributes and to improve reflection events in noisy data with a very low fold. We reveal that VFSA is efficient in reaching the optimal coherence value with the lowest computational costs, and that DE is effective and reliable in reaching the optimal coherence for determining the best five searched-for attributes. Regardless of the differences, the FO-CRS stack produces enhanced and regularized high-quality CO sections using both global optimization methods.

Traveltime approximation for converted waves in elastic orthorhombic media

The moveout approximations are commonly used in seismic data processing such as velocity analysis, modeling, and time migration. The anisotropic effect is very obvious for a converted wave when estimating the physical and processing parameters from the real data. To approximate the traveltime in an elastic orthorhombic (ORT) medium, we defined an explicit rational-form approximation for the traveltime of the converted [Formula: see text]-, [Formula: see text]-, and [Formula: see text]-waves. To obtain the expression of the coefficients, the Taylor-series approximation is applied in the corresponding vertical slowness for three pure-wave modes. By using the effective model parameters for [Formula: see text]-, [Formula: see text]-, and [Formula: see text]-waves, the coefficients in the converted-wave traveltime approximation can be represented by the anisotropy parameters defined in the elastic ORT model. The accuracy in the converted-wave traveltime for three ORT models is illustrated in numerical examples. One can see from the results that, for converted [Formula: see text]- and [Formula: see text]-waves, our rational-form approximation is very accurate regardless of the tested ORT model. For a converted [Formula: see text]-wave, due to the existence of cusps, triplications, and shear singularities, the error is relatively larger compared with PS-waves.

S‐wave in 2D acoustic transversely isotropic media with a tilted symmetry axis

ABSTRACTIn an acoustic transversely isotropic medium, there are two waves that propagate. One is the P‐wave and another one is the S‐wave (also known as S‐wave artefact). This paper is devoted to analyse the S‐wave in two‐dimensional acoustic transversely isotropic media with a tilted symmetry axis. We derive the S‐wave slowness surface and traveltime function in a homogeneous acoustic transversely isotropic medium with a tilted symmetry axis. The S‐wave traveltime approximations in acoustic transversely isotropic media with a tilted symmetry axis can be mapped from the counterparts for acoustic transversely isotropic media with a vertical symmetry axis. We consider a layered two‐dimensional acoustic transversely isotropic medium with a tilted symmetry axis to analyse the S‐wave moveout. We also illustrate the behaviour of the moveout for reflected S‐wave and converted waves.

COMPARISON OF GLOBAL SEARCH OPTIMIZATION ALGORITHMS TO PERFORM THE VELOCITY ANALYSIS WITH A CONVERTED WAVE MOVEOUT EQUATION

Even with previous works having studied about the accuracy and objective function of several nonhyperbolic multiparametric travel-time approximations for velocity analysis, they lack tests concerning different optimization algorithms and how they influence the accuracy and processing time. Once many approximations were tested and found the multimodal one which presented the best accuracy results, it is possible to perform a velocity analysis with different global search optimization algorithms. The minimization of the curve calculated with the converted wave moveout equation to the observed curve can be done for each optimization algorithm selected in this work. The travel-time curves tested here are the PP and PS reflection events coming from the interface of the top of an offshore ultra-deep reservoir. After the inversion routine have been performed, it is possible to define the processing time and the accuracy of each optimization algorithm for this kind of problem.

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.

LOCAL SEARCH OPTIMIZATION: A COMPARISON OF ALGORITHMS FOR NONHYPERBOLIC TRAVEL-TIME ANALYSIS

In the last decade, many works compared nonhyperbolic multiparametric travel-time approximations to perform velocity analysis. In these works, some analyses were accomplished, such as accuracy analysis and objective function analysis. However, no previous works compared the optimization algorithms to perform the inversion procedure concerning the processing time and the accuracy of each algorithm. As the shifted hyperbola showed the best results among the unimodal approximations in previous works, it was selected to be used in a comparison with five local search optimization algorithms. Each algorithm was compared concerning the accuracy by the minimization of the calculated curve to the observed curve. The travel-time curves tested here are conventional (PP) and converted wave (PS) reflection events from an offshore model. With this set of tests, it is possible to define which optimization algorithm presents the most reliable result when used with the shifted hyperbola equation concerning the processing time and the accuracy.