Normal Moveout Velocity Research Articles

Asymptotic linearized inversion for acoustic variable-tilt transversely isotropic media through elliptic perturbation analysis

The linearized inverse scattering problem for acoustic variable-tilt transversely isotropic (TTI) media is studied. In depicting the acoustic anisotropy behavior of the TTI medium with the P-wave normal moveout velocity (NMO) Vn, density ρ, and anisotropic parameters δ and η, a single-scattering procedure of NMO pressure wavefields is developed through elliptical perturbation analysis. By high-frequency approximation, the singly scattered pressure field is naturally related with an acoustic TTI generalized Radon transform (GRT), in which the TTI scattering patterns are explicitly expressed. We asymptotically invert for the multiple material parameters of an acoustic TTI medium via constructing the GRT backprojection operator. The effectiveness of the proposed approach is demonstrated on the layered model, and its superiority over the traditional acoustic isotropic and VTI GRT inversion methods is illustrated on the overthrust model.

Low-rank-based residual statics estimation and correction

Surface consistency forms the basis for short-wavelength statics estimation. When raypaths in the near surface diverge from a normal incidence or when the normal moveout (NMO) velocity is inaccurate, surface-consistent methods may fail to estimate accurate statics. Existing nonsurface-consistent techniques can be prone to errors due to the need to construct pilot traces or pick horizons while imposing additional computational costs. To overcome these limitations and correct for the surface- and nonsurface-consistent statics, we have developed a low-rank-based residual statics (LR-ReS) estimation and correction framework. The method makes use of the redundant nature of seismic data by using its low-rank structure in the midpoint-offset-frequency domain. Due to the near-surface effect, the low-rank structure is destroyed. Therefore, we estimate the statics by means of low-rank approximation and crosscorrelation. To alleviate the need for accurate rank selection for low-rank approximation and improved statics estimation, we implement the method in an iterative and multiscale fashion. Because the low-rank approximation deteriorates at high frequencies, we use its better performance at low frequencies and exploit the common statics among the different frequency bands. The LR-ReS estimation and correction can be applied to data without an NMO correction, which makes statics estimation independent of the NMO velocity errors. Consequently, it can reduce the multiple iterations of the NMO velocity estimation and short-wavelength statics correction commonly needed for conventional methods to improve their performance. Moreover, the LR-ReS estimation does not require windowing of a noise-free area containing aligned primaries or mute to avoid the NMO stretch effect, which enables statics correction of the wavefield of all offsets. To evaluate the performance of our method, we apply it to simulated data and a challenging field data set affected by complex weathering layers and noise, which indicate a substantial improvement compared with conventional short-wavelength statics correction.

Multiparameter Acoustic Inversion for Variable-Tilt Transversely Isotropic Media With Generalized Radon Transform

The pseudoacoustic approximations are commonly used for migration and inversion in transversely isotropic (TI) media, as they accurately characterize the P-wave propagation and are simpler than their elastic counterparts, resulting in computational savings. This paper is devoted to presenting an approach for generalized Radon transform (GRT) migration and inversion in acoustic TI media with a tilted symmetry axis (TTI). In parameterizing an acoustic TTI medium with the P-wave normal moveout velocity (NMO) <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v_n</i> , Thomsen’s parameter δ, and anelliptic parameter η, a concise single-scattering integral for NMO pressure is obtained by perturbing the TTI medium from a background non-elliptically anisotropic medium. It results in explicitly representing the perturbation scattering patterns of each parameter ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v_n</i> , δ, η), helping us understand the scattering angular influence of the perturbed parameters. The application of GRT on this scattering integral allows a direct construction of the acoustic TTI inversion operator. Numerical examples verify the effectiveness of the proposed acoustic TTI GRT inversion method and show its considerably good performance in the presence of steeply dipping anisotropic layering.

Continuous Automatic Estimation of Volumetric Water Content Profile During Infiltration Using Sparse Multi-Offset GPR Data

Ground-penetrating radar (GPR) is a non-destructive and non-invasive geophysical survey method that has been used to characterize soil volumetric water content (VWC) dynamics. An array antenna GPR system was used to collect nearly seamless, time-lapse multi-offset GPR data during an in-situ infiltration test on sand dunes with limited traces. Because the data volume was significant, an approach was utilized to automatically determine electromagnetic wave velocities from sparse common midpoint (CMP) data using standard velocity analysis, such as semblance analysis. The objective of this study was to develop a methodology that allows one to automatically perform velocity analysis by interpolating sparse CMP data obtained with the array GPR system. The proposed method determined the optimal normal moveout velocity values and the removal range of the F-K zone pass filter that minimized errors between the original and interpolated CMP data using leave-one-out cross-validation (LOOCV). After interpolating the sparse CMP data with the F-K zone pass filter, semblance analysis was used to determine the time-lapse velocity structure of the soil profile during water infiltration. The velocity data were converted to VWC data based on the Topp equation, which relates the soil VWC to the soil dielectric constant. The proposed method was tested using CMP data obtained via numerical simulation and experiments. The VWC profile from the proposed approach matched well with the independently observed VWC profiles obtained from an invasive probe-type soil moisture sensor.

3-D pre-stack diffraction separation by extending the PWD method with parametrized local slope

SUMMARY Diffractions comprise the seismic response of subsurface geological discontinuities and thus can provide detailed geological information during 3-D seismic exploration. The separation of weak diffractions from specular reflections is challenging, especially when the diffractions and reflections have similar kinematical characteristics in the 3-D pre-stack case. Conventional separation methods often estimate the local slope based on optimization or the Hilbert transform, which directly depends on the distribution behaviour of seismic events and may lead to the aliasing effects of the hyperbolic reflected and diffracted slopes in the shot domain. In this study, a different method is employed: local slopes are parametrized and constrained to the normal moveout velocity and ray parameter, which can distinguish reflections and diffractions in the shot domain and enhance the stability and accuracy of local slopes. Interestingly, when the receiver line and the geological edge are coplanar, the corresponding edge diffractions and reflections exhibit extremely similar behaviour, rendering them indistinguishable. Considering this phenomenon, a 3-D pre-stack diffraction separation strategy is proposed based on the estimation of the local slopes in two orthogonal directions. Thus, the accurate local slope can be used for plane-wave destruction when separating diffractions in the 3-D pre-stack domain. Synthetic and field data applications demonstrate that the proposed separation strategy is effective and can obtain high-quality diffraction wavefields for detecting the subsurface discontinuous structure.

Impact of source and receiver distributions on imaging of dipping reflectors with passive reflection seismic interferometry

SUMMARY Passive reflection seismic interferometry (PRSI) facilitates imaging the subsurface structure using passive sources but according to the general theory, the target needs to be illuminated equally from all directions to obtain a kinematically correct result. In practice, this requirement is almost impossible to meet which can introduce artefacts into the PRSI results. Our study was motivated by an unsuccessful attempt to image a couple of known dipping reflectors by applying PRSI to a data set of local microearthquakes clustering around the glacially triggered Burträsk fault. Dipping reflectors are a special case since they introduce a directionality into the seismic-interferometry problem that makes the results especially sensitive to the source azimuths. To investigate which source distributions are favourable in such a case and to study the range of artefacts occurring, we analyse a number of acoustic and elastic synthetic data sets calculated using a simple model of a dipping fault. Our results show that the main contribution for imaging such a fault with PRSI comes from sources in the hangingwall whereas contributions from the footwall are often weak and kinematically incorrect. The type and position of the occurring artefacts depend upon both the source azimuth and the type of modelling. In the acoustic case, the main artefact is a gently dipping reflection caused by insufficient cancellation of the direct reflection at the fault. In the elastic case, the artefacts are dominated by a set of both gently and steeply dipping reflections related to P–S converted waves. These artefacts are present even for ideal illumination due to the use of source records containing both P- and S-wave contributions. During interpretation, it is essential to be able to distinguish between physically meaningful reflections and artefacts. We found that both acoustic and elastic artefacts stack best at lower than expected normal moveout velocities. If data quality is insufficient for velocity analysis, our results can serve as a reference point for the interpretation of dipping features in PRSI images.

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.

Wavelet estimation and nonstretching NMO correction

Normal-moveout (NMO) correction is an important step applied to common-midpoint (CMP) gathers for subsequent stacking. However, conventional NMO correction methods often suffer from the problem of NMO stretching, which nonlinearly increases with offsets and decreases with zero-offset traveltime. The NMO stretching can be quantified by frequency distortion, so stretching is confined mainly to large offsets and shallow times. To solve this problem, we have proposed a wavelet-based method with the following four steps. First, we estimate a wavelet from the CMP gather by using the NMO stretching appearing in the conventional NMO correction. Second, we deconvolve the original CMP gather based on the estimated wavelet. This step of removing the wavelet from the CMP gather is helpful for the next steps of NMO velocity scan and NMO correction. Third, we apply an improved NMO correction to the deconvolved CMP gather and obtain flattened reflectivities. Finally, we convolve the flattened and deconvolved gather with the estimated wavelet back to obtain an NMO-corrected gather without stretching artifacts. In our method, by using a deconvolved CMP gather, we are able to calculate a high-resolution semblance velocity spectrum that benefits from the NMO velocity picking. In addition, applying the NMO correction to the deconvolved CMP gather, instead of the original gather, is helpful in reducing the NMO stretching related to the wavelet distortion. Tests on synthetic and field data find that our new NMO correction method can estimate an accurate seismic wavelet and obtain an NMO-corrected gather without NMO stretching. Reducing the NMO stretching can significantly improve the resolution of shallow layers at far offsets and preserve the spectral bandwidth.

Automatic Velocity Picking Using a Multi-Information Fusion Deep Semantic Segmentation Network

Velocity picking, a critical step in seismic data processing, has been studied for decades. Although manual picking can produce accurate normal moveout (NMO) velocities from the velocity spectra of prestack gathers, it is time-consuming and becomes infeasible with the emergence of large amount of seismic data. Numerous automatic velocity picking methods have thus been developed. In recent years, deep learning (DL) methods have produced good results on the seismic data with medium and high signal-to-noise ratios (SNR). Unfortunately, it still lacks a picking method to automatically generate accurate velocities in the situations of low SNR. In this paper, we propose a multi-information fusion network (MIFN) to estimate stacking velocity from the fusion information of velocity spectra and stack gather segments (SGS). In particular, we transform the velocity picking problem into a semantic segmentation problem based on the velocity spectrum images. Meanwhile, the information provided by SGS is used as a prior in the network to assist segmentation. The experimental results on two field datasets show that the picking results of MIFN are stable and accurate for the scenarios with medium and high SNR, and it also performs well in low SNR scenarios.

Born Scattering Integral, Scattering Radiation Pattern, and Generalized Radon Transform Inversion in Acoustic Tilted Transversely Isotropic Media

Although the pseudo-acoustic wave equation has good accuracy in characterizing anisotropic wave propagation and obtaining interpretable seismic images, a high-precision multiparameter inversion accounting for anisotropy from compressional wavefields is still confronted with challenges, even in the simple transversely isotropic (TI) case, due to the complicated relationship between anisotropic properties and pressure data. To reduce difficulty in correctly inverting surface compressional data in anisotropic media, an appropriate parameterization for inversion is necessary. For acoustic TI media with a tilted symmetry axis (TTI), we describe the pseudo-acoustic TTI equations with the P-wave normal moveout velocity vn and anisotropic parameters η and δ, and aim to invert for this parameterization by the scattering integral method. Using perturbation theory in formulating the integral solution of the singly scattered pressure wavefield allows to acquire a scattering radiation pattern that explicitly illustrates the angular effect (including migration dip and scattering angles) of the TTI perturbation parameters, in which perturbations whether in the wavefield or anisotropic parameters are from the elliptical anisotropic background medium. Taking advantage of a ray-theoretical approximation to the background Green’s function, we can establish a relationship between the scattering integral and a form of TTI generalized Radon transform (GRT). As a result, we develop an acoustic TTI pseudoinverse GRT operator for estimating the corresponding perturbation parameters. Numerical tests on two simple models and a part of BP 2007 anisotropic benchmark model verify the effectiveness of the presented acoustic TTI GRT inverison/migration method and show its evident advantages over the conventional acoustic isotropic and TI with a vertical symmetry axis (VTI) approaches.

Multi-Parameter True-Amplitude Generalized Radon Transform Inversion for Acoustic Transversely Isotropic Media With a Vertical Symmetry Axis

In anisotropic media, the compressional wave scattering phenomena is governed by several parameters and can be simulated using a good kinematic approximation. How to retrieve anisotropic properties from recorded seismic data free of shear waves for exploration geophysics is interesting and challenging. We present an approach to infer the material properties in acoustic transversely isotropic (TI) media with a vertical axis of symmetry (VTI) described by a combination of the normal-moveout velocity <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$v_{n}$ </tex-math></inline-formula> and anisotropic parameters <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\eta $ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula> . The method we consider is based on the true-amplitude generalized Radon transform (GRT) inversion strategy. Because the scattering integral is at the kernel of the inversion engine, we start our investigation from a fourth-order pseudo-acoustic VTI equation instead of the conventionally coupled system of second-order wave equations. With the single-scattering approximation and high-frequency asymptotic analysis, the integral representation of P-wave scattered wavefield is incorporated into a weighted GRT operator that contains VTI scattering patterns of each parameter perturbation, which leads the way in constructing an acoustic VTI amplitude-preserving GRT pseudo-inverse operator. We describe an appropriate survey design by shooting a fan of rays from the target area toward the acquisition system, which is necessary when calculating the pseudo-inverse operator. Numerical test results from 2-D synthetic data verify the effectiveness of our method.

Prestack diffraction separation by parameterizing the reflection local slope

Diffracted waves provide an opportunity to detect small-scale subsurface structures because they give wide illumination direction of geologic discontinuities, such as faults, pinch outs, and collapsed columns. However, separating diffracted waves is challenging because diffracted waves have greater geometric amplitude losses and are generally weaker than reflections. To retain more diffracted waves, we have developed a prestack diffraction separation method based on the local slope pattern and plane-wave destruction (PWD) method. In general, it is difficult to distinguish between hyperbolic reflections and hyperbolic diffractions using data-driven local slope estimation in the shot domain. Therefore, we transfer slope estimation in the shot domain to velocity analysis in the common-midpoint domain and ray parameter calculation in the stack domain. The connection between local slope and the normal moveout velocity and the surface-ray parameter is known, which provides a novel approach for estimating the local slope of hyperbolic reflected waves in the shot domain. The estimated slope can provide an exact slope-based operator for the PWD method, thus allowing the PWD to separate diffracted waves from reflected waves in the shot domain. Synthetic and field data tests demonstrate the feasibility and effectiveness of our prestack diffraction separation method.

Feasibility study of fracture interpretation using multicomponent seismic data: SEAM Phase II Barrett model

Several P-wave azimuthal anisotropy studies have been conducted for the SEG Advanced Modeling Program Phase II Barrett model data. However, these analyses provide fracture property estimation that is inconsistent with the actual model properties. Therefore, we perform a feasibility study to understand the influence of the overburden and reservoir properties, and processing and inversion steps, which together determine the success of the fracture interpretation from seismic data. The 1D model properties (orthorhombic for overburden and reservoir) are first extracted from the actual Barrett model properties at two locations. Anisotropic prestack reflectivity modeling exposes the true orthorhombic response of the 1D medium in the form of common-offset and common-azimuth gathers. The true anisotropic response is obscured in the Barrett data (generated by finite-element modeling) due to the mild lateral velocity variations and orthorhombic anisotropy in the overburden. We then expose the reservoir anisotropic response using an isotropic overburden in the reflectivity modeling. This indicates that the P-wave velocity variation with azimuth (VVAZ) responses generated by the reservoir itself are weak, which leads to an unstable VVAZ inversion to estimate the interval normal moveout (NMO) velocity anisotropy. The reservoir thickness (125 m or 65 ms of two-way traveltime) or NMO velocity anisotropy (6%–7%) needs to be at least doubled to obtain a stable VVAZ inversion. Anisotropic geometric-spreading correction improves the amplitude variation with azimuth inversion results when reflectivity modeling models the orthorhombic overburden. The converted wave (C-wave) has a stronger VVAZ response compared to that of the P-wave. We suggest that the C-wave data could be useful to constrain fracture interpretation in the Barrett model. We conclude that the results of previous studies are due to the combination of the residual influence of overburden after processing and imaging and the weak anisotropy responses from the reservoir.

Elastic FWI for orthorhombic media with lithologic constraints applied via machine learning

Full-waveform inversion (FWI) of 3D wide-azimuth data for elastic orthorhombic media suffers from parameter trade-offs which cannot be overcome without constraining the model-updating procedure. We present an FWI methodology that incorporates geologic constraints to reduce the inversion nonlinearity and increase the resolution of parameter estimation for orthorhombic models. These constraints are obtained from well logs, which can provide rock-physics relationships for different geologic facies. Because the locations of the available well logs are usually sparse, a supervised machine-learning (ML) algorithm (Support Vector Machine) is employed to account for lateral heterogeneity in building the lithologic constraints. The advantages of the facies-based FWI are demonstrated on the modified SEG-EAGE 3D overthrust model, which is made orthorhombic with the symmetry planes that coincide with the Cartesian coordinate planes. We employ a velocity-based parameterization, whose suitability for FWI was studied using the radiation-pattern analysis. Application of the facies-based constraints substantially increases the resolution of the P- and S-wave vertical velocities ([Formula: see text], [Formula: see text], and [Formula: see text]) and, therefore, of the depth scale of the model. Improvements are also observed for the P-wave horizontal and normal-moveout velocities ([Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]) and the S-wave horizontal velocity [Formula: see text]. However, the velocity [Formula: see text] that depends on Tsvankin’s parameter [Formula: see text] defined in the horizontal plane is not well recovered from the surface data. On the whole, the developed algorithm achieves a much higher spatial resolution compared to unconstrained FWI, even in the absence of recorded frequencies below 2 Hz.

Linearized scattering problem in acoustic transversely isotropic media with a vertical symmetry axis

The ray-based generalized Radon transform (GRT) inversion/migration strategy is studied in this paper for acoustic transversely isotropic media with a vertical symmetry axis (VTI media) represented by the P-wave normal moveout velocity, Thomsen parameter δ, anelliptic parameter η, and density. The multi-parameter GRT inversion solutions for multi-shot/multi-offset acquisition geometries are derived from the single-scattering approximation of the wavefields and an elliptically anisotropic background. Results from classical two-dimensional numerical tests show the true amplitude properties of the GRT backprojection operators.

Estimation of P‐wave anisotropy parameters from 3D vertical seismic profile with distributed acoustic sensors and geophones for seismic imaging in the CO2CRC Otway Project

ABSTRACTThe quality and accuracy of a seismic image can be significantly affected by anisotropy. When anisotropy is significant, neglecting it can lead to smearing of the image, dispositioning of reflections and distortion of amplitudes. Thus, it is imperative to estimate anisotropy and take it into account, especially for long‐offset or wide‐azimuthal seismic data. This study estimates P‐wave anisotropy parameters at the Otway site (Victoria, Australia). A wide range of available offsets acquired within the CO2CRC Otway Project provides a unique opportunity for anisotropy estimation from 3D vertical seismic profile data. We estimate P‐wave anisotropy from direct‐wave arrivals in vertical seismic profile data acquired with geophones and distributed acoustic sensors installed in two wells. Data analysis reveals seismic anisotropy of the subsurface with a significant presence of both polar and azimuthal anisotropy. We estimate key parameters such as normal moveout velocity, azimuthal ellipticity, polar anellipticity and fast‐velocity azimuth by nonlinear fitting of the analytical travel‐time approximation to observed direct P‐wave travel times from 3D vertical seismic profile data. Obtained anisotropy parameters show that polar anellipticity remains almost constant and equal to 0.1 for the whole depth range, while azimuthal anisotropy changes significantly with depth: it is relatively weak in the shallow section then increases significantly below a depth of 600 m. The fast‐velocity azimuth coincides with the orientation of the maximum horizontal stress in the study area. The results show that despite their low sensitivity for large offsets, the distributed acoustic sensor receivers provide sufficient information on anisotropic travel times over the entire length of the well, which is critical for reflection imaging from both vertical seismic profile and surface seismic data.

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.

On parameterization in monoclinic media with a horizontal symmetry plane

I have derived accurate anisotropy parameters for a monoclinic anisotropy model with a horizontal symmetry plane based on normal moveout (NMO) ellipses for P-, S1-, S2-, and converted waves. The NMO velocity ellipse is also defined for all types of converted waves. The parameters are defined in the phase domain and compared with existing approximate monoclinic anisotropy parameters. These parameters are evaluated for two benchmark models consisting of two nonorthogonal fracture sets embedded into a transversely isotropic medium with a vertical symmetry axis. The dependence of monoclinic parameters on the azimuth angle between the fracture sets is analyzed. Being linearized with respect to fracture weaknesses, the monoclinic anisotropy parameters can be decomposed into sine functions of double and quartic azimuth angle between the fracture sets with the weights given by the stiffness coefficients of the background model. The discrimination between the fracture parameters computed from a given set of monoclinic parameters is dependent on the background model and controlled by the azimuth angle between the fracture sets.

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.

Velocity and event‐slope analysis using a model‐based common‐diffraction‐surface stack operator

ABSTRACTOver the last two decades, scientists have introduced many ways to improve normal moveout velocity analysis by optimizing the resolution of the moveout velocity spectrum, a graph that displays a coherence value for every tested velocity traveltime pair. Almost all of these methods have failed to enhance resolution when faced with low‐fold common‐midpoint gathers, which might be caused by natural barriers or man‐made obstacles. Another problem is that many approaches are derived from very simple model assumptions that quickly break down for complex structures and do not provide enough model flexibility for an iterative and interactive velocity analysis. In this paper, we present a new velocity analysis method based on the model‐based common‐diffraction‐surface stack operator and apply it to two synthetic data sets, one with locally sparse common‐midpoint coverage and one with a laterally variable complex geological structure. We generate velocity spectra by calculating the semblance along spatial operators obtained for all possible emergence angles and an entire range of velocity models. Comparing the resolution of such velocity spectra with those obtained with the classical normal moveout velocity analysis shows, in both two analysed cases, that the stacking velocity can be estimated much more precisely. The reasons for this are that the event dip is handled independently from the velocity and that the semblance is obtained, for each zero‐offset sample, over a group of neighbouring common‐midpoint gathers instead of just one.