VTI介质P波非双曲时差分析

具有垂直对称轴的横向各向同性介质模型(VTI)是目前各向异性理论研究和多波多分量地震资料叠前成像处理中最常用的一种各向异性模型.VTI介质中反射 P波时距曲线一般不再是双曲线.基于不同的相速度近似公式会得到不同的时距关系式.文中对几种典型的非双曲时距曲线与射线追踪得到的准确时距曲线在不同各向异性强度下进行了对比，结果表明Muir等和Stovas等提出的非双曲时距公式由于过高地考虑了横波垂直速度的影响与精确的时距曲线有很大偏差；Tsvankin等提出的弱各向异性非双曲时距公式在ε-δ<0时误差增大；Alkhalifah等提出的非双曲时距公式在大炮检距任意各向异性强度下都具有较高的精度，适于在实际资料处理中应用.     

Accounting for velocity anisotropy in seismic traveltime tomography: a case study from the investigation of the foundations of a Byzantine monumental building

We estimate velocity anisotropy factors from seismic traveltime tomographic data and apply a correction for anisotropy in the inversion procedure to test possible improvements on the traveltime fit and the quality of the resulting tomographic images. We applied the anisotropy correction on a traveltime data set obtained from the investigation of the foundation structure of a monumental building: a Byzantine church from the 11th century AD, in Athens, Greece. Vertical transverse isotropy is represented by one axis of symmetry and one anisotropy magnitude for the entire tomographic inversion grid. We choose the vertical direction for the symmetry axis by analysing the available data set and taking into account information on the character of the foundations of the church from the literature and past excavations. The anisotropy magnitude is determined by testing a series of values of anisotropy and examining their effect on the tomographic inversion results. The best traveltime fit and image quality are obtained with an anisotropy value (V_max/V_min) of 1.6, restricted to the high velocity structures in the subsurface. We believe that this anisotropy value, which is significantly higher than the usual values reported for near‐surface geological material, is related to the fabric of the church foundations, due to the shape of the individual stone blocks and the layout of the stonework. Inversion results obtained with the correction for anisotropy indicate that both the traveltime fit and the image quality are improved, providing an enhanced reconstruction of the velocity field, especially for the high‐velocity features. Based on this enhanced and more reliable reconstruction of velocity distribution, an improved image of the subsurface material character was made possible. In particular, the pattern and state of the church foundations and possible weak ground material areas were revealed more clearly. This improved subsurface knowledge may assist in a better design of restoration measures for monumental buildings such as Byzantine churches.     

Anisotropic migration velocity analysis: Application to a data set from West Africa

Although it is widely recognized that anisotropy can have a significant influence on the focusing and positioning of migrated reflection events, conventional depth imaging methods still operate with isotropic velocity fields. Here, we present an application of a 2D migration velocity analysis (MVA) algorithm, designed for factorized v(x, z) VTI (transversely isotropic with a vertical symmetry axis) media, to an offshore data set from West Africa. By approximating the subsurface with factorized VTI blocks, it is possible to decouple the spatial variations in the vertical velocity from the anisotropic parameters with minimal a priori information. Since our method accounts for lateral velocity variation, it produces more accurate estimates of the anisotropic parameters than those previously obtained with time‐domain techniques. The values of the anellipticity parameter η found for the massive shales exceed 0.2, which confirms that ignoring anisotropy in the study area can lead to substantial imaging distortions, such as mis‐stacking and mispositioning of dipping events. While some of these distortions can be removed by using anisotropic time processing, further marked improvement in image quality is achieved by prestack depth migration with the estimated factorized VTI model. In particular, many fault planes, including antithetic faults in the shallow part of the section, are better focused by the anisotropic depth‐migration algorithm and appear more continuous. Anisotropic depth migration facilitates structural interpretation by eliminating false dips at the bottom of the section and improving the images of a number of gently dipping features. One of the main difficulties in anisotropic MVA is the need to use a priori information for constraining the vertical velocity. In this case study, we successfully reconstructed the time–depth curve from reflection data by assuming that the vertical velocity is a continuous function of depth and estimating the vertical and lateral velocity gradients in each factorized block. If the subsurface contains strong boundaries with jumps in velocity, knowledge of the vertical velocity at a single point in a layer is sufficient for our algorithm to determine all relevant layer parameters.     

Non‐uniqueness with refraction inversion – a syncline model study

Non‐uniqueness occurs with the 1D parametrization of refraction traveltime graphs in the vertical dimension and with the 2D lateral resolution of individual layers in the horizontal dimension. The most common source of non‐uniqueness is the inversion algorithm used to generate the starting model. This study applies 1D, 1.5D and 2D inversion algorithms to traveltime data for a syncline (2D) model, in order to generate starting models for wave path eikonal traveltime tomography. The 1D tau‐p algorithm produced a tomogram with an anticline rather than a syncline and an artefact with a high seismic velocity. The 2D generalized reciprocal method generated tomograms that accurately reproduced the syncline, together with narrow regions at the thalweg with seismic velocities that are less than and greater than the true seismic velocities as well as the true values. It is concluded that 2D inversion algorithms, which explicitly identify forward and reverse traveltime data, are required to generate useful starting models in the near‐surface where irregular refractors are common. The most likely tomogram can be selected as either the simplest model or with a priori information, such as head wave amplitudes. The determination of vertical velocity functions within individual layers is also subject to non‐uniqueness. Depths computed with vertical velocity gradients, which are the default with many tomography programs, are generally 50% greater than those computed with constant velocities for the same traveltime data. The average vertical velocity provides a more accurate measure of depth estimates, where it can be derived. Non‐uniqueness is a fundamental reality with the inversion of all near‐surface seismic refraction data. Unless specific measures are taken to explicitly address non‐uniqueness, then the production of a single refraction tomogram, which fits the traveltime data to sufficient accuracy, does not necessarily demonstrate that the result is either ‘correct’ or the most probable.     

正交各向异性介质P波走时分析及Thomsen参数反演

对于包含有垂向裂缝的横向各向同性地层或含有多组正交裂缝的各向同性地层，正交各向异性介质模型是最简单的与实际地层相符的方位各向异性模型.本文对单层水平反射界面正交各向异性模型采用射线追踪法计算了全方位角变化的P波走时，时距曲线表现出强方位各向异性.采用小生境遗传算法，对三条成一定角度的测线的走时信息进行速度和各向异性参数反演.模型算例表明，此方法可以得到高精度的裂缝方位角、P波垂直速度和较高精度的Thomsen各向异性参数.     

Depth migration anisotropy analysis in the time domain

In areas of complex geology such as the Canadian Foothills, the effects of anisotropy are apparent in seismic data and estimation of anisotropic parameters for use in seismic imaging is not a trivial task. Here we explore the applicability of common‐focus point (CFP)‐based velocity analysis to estimate anisotropic parameters for the variably tilted shale thrust sheet in the Canadian Foothills model. To avoid the inherent velocity‐depth ambiguity, we assume that the elastic properties of thrust‐sheet with respect to transverse isotropy symmetry axis are homogeneous, the reflector below the thrust‐sheet is flat, and that the anisotropy is weak. In our CFP approach to velocity analysis, for a poorly imaged reflection point, a traveltime residual is obtained as the time difference between the focusing operator for an assumed subsurface velocity model and the corresponding CFP response obtained from the reflection data. We assume that this residual is due to unknown values for anisotropy, and we perform an iterative linear inversion to obtain new model parameters that minimize the residuals. Migration of the data using parameters obtained from our inversion results in a correctly positioned and better focused reflector below the thrust sheet. For traveltime computation we use a brute force mapping scheme that takes into account weakly tilted transverse isotropy media. For inversion, the problem is set up as a generalized Newton's equation where traveltime error (differential time shift) is linearly dependent on the parameter updates. The iterative updates of parameters are obtained by a least‐squares solution of Newton's equations. The significance of this work lies in its applicability to areas where transverse isotropy layers are heterogeneous laterally, and where transverse isotropy layers are overlain by complex structures that preclude a moveout curve fitting.     

Moveout approximation for horizontal transversely isotropic and vertical transversely isotropic layered medium. Part II: effective model‡

We use residual moveouts measured along continuous full azimuth reflection angle gathers, in order to obtain effective horizontal transversely isotropic model parameters. The angle gathers are generated through a special angle domain imaging system, for a wide range of reflection angles and full range of phase velocity azimuths. The estimation of the effective model parameters is performed in two stages. First, the background horizontal transversely isotropic (HTI)/vertical transversely isotropic (VTI) layered model is used, along with the values of reflection angles, for converting the measured residual moveouts (or traveltime errors) into azimuthally dependent normal moveout (NMO) velocities. Then we apply a digital Fourier transform to convert the NMO velocities into azimuthal wavenumber domain, in order to obtain the effective HTI model parameters: vertical time, vertical compression velocity, Thomsen parameter delta and the azimuth of the medium axis of symmetry. The method also provides a reliability criterion of the HTI assumption. The criterion shows whether the medium possesses the HTI type of symmetry, or whether the azimuthal dependence of the residual traveltime indicates to a more complex azimuthal anisotropy. The effective model used in this approach is defined for a 1D structure with a set of HTI, VTI and isotropic layers (with at least one HTI layer). We describe and analyse the reduction of a multi‐layer structure into an equivalent effective HTI model. The equivalent model yields the same NMO velocity and the same offset azimuth on the Earth's surface as the original layered structure, for any azimuth of the phase velocity. The effective model approximates the kinematics of an HTI/VTI layered structure using only a few parameters. Under the hyperbolic approximation, the proposed effective model is exact.     

Traveltime approximation in vertical transversely isotropic layered media

The 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.     

3D P‐wave traveltime computation in transversely isotropic media using layer‐by‐layer wavefront marching

Subsurface rocks (e.g. shale) may induce seismic anisotropy, such as transverse isotropy. Traveltime computation is an essential component of depth imaging and tomography in transversely isotropic media. It is natural to compute the traveltime using the wavefront marching method. However, tracking the 3D wavefront is expensive, especially in anisotropic media. Besides, the wavefront marching method usually computes the traveltime using the eikonal equation. However, the anisotropic eikonal equation is highly non‐linear and it is challenging to solve. To address these issues, we present a layer‐by‐layer wavefront marching method to compute the P‐wave traveltime in 3D transversely isotropic media. To simplify the wavefront tracking, it uses the traveltime of the previous depth as the boundary condition to compute that of the next depth based on the wavefront marching. A strategy of traveltime computation is designed to guarantee the causality of wave propagation. To avoid solving the non‐linear eikonal equation, it updates traveltime along the expanding wavefront by Fermat's principle. To compute the traveltime using Fermat's principle, an approximate group velocity with high accuracy in transversely isotropic media is adopted to describe the ray propagation. Numerical examples on 3D vertical transverse isotropy and tilted transverse isotropy models show that the proposed method computes the traveltime with high accuracy. It can find applications in modelling and depth migration.     

Effective wavefield extrapolation in anisotropic media: accounting for resolvable anisotropy

Spectral methods provide artefact‐free and generally dispersion‐free wavefield extrapolation in anisotropic media. Their apparent weakness is in accessing the medium‐inhomogeneity information in an efficient manner. This is usually handled through a velocity‐weighted summation (interpolation) of representative constant‐velocity extrapolated wavefields, with the number of these extrapolations controlled by the effective rank of the original mixed‐domain operator or, more specifically, by the complexity of the velocity model. Conversely, with pseudo‐spectral methods, because only the space derivatives are handled in the wavenumber domain, we obtain relatively efficient access to the inhomogeneity in isotropic media, but we often resort to weak approximations to handle the anisotropy efficiently. Utilizing perturbation theory, I isolate the contribution of anisotropy to the wavefield extrapolation process. This allows us to factorize as much of the inhomogeneity in the anisotropic parameters as possible out of the spectral implementation, yielding effectively a pseudo‐spectral formulation. This is particularly true if the inhomogeneity of the dimensionless anisotropic parameters are mild compared with the velocity (i.e., factorized anisotropic media). I improve on the accuracy by using the Shanks transformation to incorporate a denominator in the expansion that predicts the higher‐order omitted terms; thus, we deal with fewer terms for a high level of accuracy. In fact, when we use this new separation‐based implementation, the anisotropy correction to the extrapolation can be applied separately as a residual operation, which provides a tool for anisotropic parameter sensitivity analysis. The accuracy of the approximation is high, as demonstrated in a complex tilted transversely isotropic model.     

Traveltime computation with the linearized eikonal equation for anisotropic media

A linearized eikonal equation is developed for transversely isotropic (TI) media with a vertical symmetry axis (VTI). It is linear with respect to perturbations in the horizontal velocity or the anisotropy parameter η. An iterative linearization of the eikonal equation is used as the basis for an algorithm of finite-difference traveltime computations. A practical implementation of this iterative technique is to start with a background model that consists of an elliptically anisotropic, inhomogeneous medium, since traveltimes for this type of medium can be calculated efficiently using eikonal solvers, such as the fast marching method. This constrains the perturbation to changes in the anisotropy parameter η (the parameter most responsible for imaging improvements in anisotropic media). The iterative implementation includes repetitive calculation of η from traveltimes, which is then used to evaluate the perturbation needed for the next round of traveltime calculations using the linearized eikonal equation. Unlike isotropic media, interpolation is needed to estimate η in areas where the traveltime field is independent of η, such as areas where the wave propagates vertically.
Typically, two to three iterations can give sufficient accuracy in traveltimes for imaging applications. The cost of each iteration is slightly less than the cost of a typical eikonal solver. However, this method will ultimately provide traveltime solutions for VTI media. The main limitation of the method is that some smoothness of the medium is required for the iterative implementation to work, especially since we evaluate derivatives of the traveltime field as part of the iterative approach. If a single perturbation is sufficient for the traveltime calculation, which may be the case for weak anisotropy, no smoothness of the medium is necessary. Numerical tests demonstrate the robustness and efficiency of this approach.     

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

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

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

Diffractions 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.     

Importance of borehole deviation surveys for monitoring of hydraulic fracturing treatments

During seismic monitoring of hydraulic fracturing treatment, it is very common to ignore the deviations of the monitoring or treatment wells from their assumed positions. For example, a well is assumed to be perfectly vertical, but in fact, it deviates from verticality. This can lead to significant errors in the observed azimuth and other parameters of the monitored fracture‐system geometry derived from microseismic event locations. For common hydraulic fracturing geometries, a 2° deviation uncertainty on the positions of the monitoring or treatment well survey can cause a more than 20° uncertainty of the inverted fracture azimuths. Furthermore, if the positions of both the injection point and the receiver array are not known accurately and the velocity model is adjusted to locate perforations on the assumed positions, several‐millisecond discrepancies between measured and modeled SH‐P traveltime differences may appear along the receiver array. These traveltime discrepancies may then be misinterpreted as an effect of anisotropy, and the use of such anisotropic model may lead to the mislocation of the detected fracture system. The uncertainty of the relative positions between the monitoring and treatment wells can have a cumulative, nonlinear effect on inverted fracture parameters. We show that incorporation of borehole deviation surveys allows reasonably accurate positioning of the microseismic events. In this study, we concentrate on the effects of horizontal uncertainties of receiver and perforation positions. Understanding them is sufficient for treatment of vertical wells, and also necessary for horizontal wells.     

Calculating complex‐valued P‐wave first‐arrival traveltimes in attenuative vertical transversely isotropic media with an irregular surface

The complex‐valued first‐arrival traveltime can be used to describe the properties of both velocity and attenuation as seismic waves propagate in attenuative elastic media. The real part of the complex‐valued traveltime corresponds to phase arrival and the imaginary part is associated with the amplitude decay due to energy absorption. The eikonal equation for attenuative vertical transversely isotropic media discretized with rectangular grids has been proven effective and precise to calculate the complex‐valued traveltime, but less accurate and efficient for irregular models. By using the perturbation method, the complex‐valued eikonal equation can be decomposed into two real‐valued equations, namely the zeroth‐ and first‐order traveltime governing equations. Here, we first present the topography‐dependent zeroth‐ and first‐order governing equations for attenuative VTI media, which are obtained by using the coordinate transformation from the Cartesian coordinates to the curvilinear coordinates. Then, we apply the Lax–Friedrichs sweeping method for solving the topography‐dependent traveltime governing equations in order to approximate the viscosity solutions, namely the real and imaginary parts of the complex‐valued traveltime. Several numerical tests demonstrate that the proposed scheme is efficient and accurate in calculating the complex‐valued P‐wave first‐arrival traveltime in attenuative VTI media with an irregular surface.     

Influence of background heterogeneity on traveltime shifts for compacting reservoirs

Compaction induced by pore‐pressure decrease inside a reservoir can be monitored by measuring traveltime shifts of reflection events on time‐lapse seismic data. Recently we introduced a perturbation‐based formalism to describe traveltime shifts caused by the 3D stress‐induced velocity field around a compacting reservoir. Application of this method to homogeneous background models showed that the offset variation of traveltime shifts is controlled primarily by the anisotropic velocity perturbations and can provide valuable information about the shear and deviatoric stresses. Here, we model and analyse traveltime shifts for compacting reservoirs whose elastic properties are different from those of the surrounding medium. For such models, the excess stress is influenced primarily by the contrast in the rigidity modulus μ across the reservoir boundaries. Synthetic examples demonstrate that a significant (25% or more) contrast in μ enhances the isotropic velocity perturbations outside the reservoir. Nevertheless, the influence of background heterogeneity is mostly confined to the reservoir and its immediate vicinity and the anisotropic velocity changes are still largely responsible for the offset dependence of traveltime shifts. If the reservoir is stiffer than the host rock, the background heterogeneity reduces anisotropic velocity perturbations inside the reservoir but increases them in the overburden. As a result, in this case, the magnitude of the offset variation of traveltime shifts is generally higher for reflections from interfaces above the reservoir. We also study compaction‐induced stress/strain and traveltime shifts for a stiff reservoir embedded in a softer layered model based on velocity profiles from the Valhall Field in the North Sea. Despite producing discontinuities in strain across medium interfaces, horizontal layering does not substantially alter the overall behaviour of traveltime shifts. The most pronounced offset variation of traveltime shifts is observed for overburden events recorded at common midpoints close to the reservoir edges. On the whole, prestack analysis of traveltime shifts should help better constrain compaction‐induced velocity perturbations in the presence of realistic background heterogeneity.     

Migration velocity analysis for tilted transversely isotropic media

Tilted transversely isotropic formations cause serious imaging distortions in active tectonic areas (e.g., fold‐and‐thrust belts) and in subsalt exploration. Here, we introduce a methodology for P‐wave prestack depth imaging in tilted transversely isotropic media that properly accounts for the tilt of the symmetry axis as well as for spatial velocity variations. For purposes of migration velocity analysis, the model is divided into blocks with constant values of the anisotropy parameters ε and δ and linearly varying symmetry‐direction velocity V_P0 controlled by the vertical (k_z) and lateral (k_x) gradients. Since determination of tilt from P‐wave data is generally unstable, the symmetry axis is kept orthogonal to the reflectors in all trial velocity models. It is also assumed that the velocity V_P0 is either known at the top of each block or remains continuous in the vertical direction. The velocity analysis algorithm estimates the velocity gradients k_z and k_x and the anisotropy parameters ε and δ in the layer‐stripping mode using a generalized version of the method introduced by Sarkar and Tsvankin for factorized transverse isotropy with a vertical symmetry axis. Synthetic tests for several models typical in exploration (a syncline, uptilted shale layers near a salt dome and a bending shale layer) confirm that if the symmetry‐axis direction is fixed and V_P0 is known, the parameters k_z, k_x, ε and δ can be resolved from reflection data. It should be emphasized that estimation of ε in tilted transversely isotropic media requires using nonhyperbolic moveout for long offsets reaching at least twice the reflector depth. We also demonstrate that application of processing algorithms designed for a vertical symmetry axis to data from tilted transversely isotropic media may lead to significant misfocusing of reflectors and errors in parameter estimation, even when the tilt is moderate (30°). The ability of our velocity analysis algorithm to separate the anisotropy parameters from the velocity gradients can be also used in lithology discrimination and geologic interpretation of seismic data in complex areas.     

各向异性TI介质qP反射波走时层析成像

地震走时层析成像是反演地层各向异性参数分布的有效方法,但是关于地震各向异性介质走时层析成像的研究并不多,其技术远远没有达到成熟的阶段.在野外数据采集时,地表反射波观测方式相对井间和垂直地震剖面观测方式的成本更低,利用qP反射波走时反演各向异性参数具有更加广泛的实用价值.本文实现的TI介质地震走时层析成像方法结合了TI介质反射波射线追踪算法、走时扰动方程和非线性共轭梯度算法,它可以对任意强度的TI介质模型进行反演,文中尝试利用qP反射波走时重建TI介质模型的参数图像.利用qP反射波对层状介质模型和块状异常体模型进行走时反演,由于qP波相速度对弹性模量参数和Thomsen参数的偏微分不同,所以可以分别反演弹性模量参数和Thomsen参数.数值模拟结果表明:利用qP反射波可以反演出TI介质模型的弹性模量参数与Thomsen参数,不同模型的走时迭代反演达到了较好的收敛效果,与各向同性介质走时反演结果相比较,各向异性介质走时反演结果具有较好的识别能力.     

First‐arrival traveltime tomography with modified total‐variation regularization

First‐arrival traveltime tomography is a robust tool for near‐surface velocity estimation. A common approach to stabilizing the ill‐posed inverse problem is to apply Tikhonov regularization to the inversion. However, the Tikhonov regularization method recovers smooth local structures while blurring the sharp features in the model solution. We present a first‐arrival traveltime tomography method with modified total‐variation regularization to preserve sharp velocity contrasts and improve the accuracy of velocity inversion. To solve the minimization problem of the new traveltime tomography method, we decouple the original optimization problem into the two following subproblems: a standard traveltime tomography problem with the traditional Tikhonov regularization and a L₂ total‐variation problem. We apply the conjugate gradient method and split‐Bregman iterative method to solve these two subproblems, respectively. Our synthetic examples show that the new method produces higher resolution models than the conventional traveltime tomography with Tikhonov regularization, and creates less artefacts than the total variation regularization method for the models with sharp interfaces. For the field data, pre‐stack time migration sections show that the modified total‐variation traveltime tomography produces a near‐surface velocity model, which makes statics corrections more accurate.