Anisotropic Mean Traveltime Curves: A Method to Estimate Anisotropic Parameters from 2D Transmission Tomographic Data

We present the mathematical deduction and properties of the mean traveltime curves for homogeneous elliptical anisotropic media. These curves generalize their isotropic counterparts which have been introduced in the past as a simple data quality analysis technique at the pre-inversion stage for 2D transmission experiments, allowing the inference of prior velocity models to gain stability at the tomographic inversion. Also, the anisotropy parameters (maximum velocity, anisotropic direction and ratio) are shown to affect the shape of these curves. The degree of asymmetry of the anisotropic mean traveltime curves (displacement of the mean time and standard deviation minima from the middle of the gathering line) is related to the direction of anisotropy which can then be visually estimated. Least squares’ fitting of the anisotropic theoretical models to their experimental counterparts is an effective method to estimate at the pre-inversion stage a macroscopic elliptical anisotropic velocity model, valid at the scale of the experiment, and able to match the experimental mean traveltime distribution. Sensitivity analysis has shown that the mean curve is less prone to errors than the standard deviation curve. Parameter identification from the standard deviation curve becomes unstable for noise levels higher than 5%; data errors produce smearing of the value of the estimated anisotropy ratio and wrong directions of anisotropy biased towards zero degrees. Also, identification from the mean traveltime curve becomes stable when the maximum velocity is well constrained. Finally, this methodology is illustrated with the application to the Grimsel data set. Performing MTC analysis is always recommended since it does not need high numerical requirements, and as shown in the sensibility analysis section, errors in data can be misinterpreted as geological anisotropies. J.L. Fernández Martínez is a visiting professor at UC Berkeley, Department of Civil and Environmental Eng., CA 94720-1710.     

用一阶扰动理论计算各向异性介质中的走时

根据各向异性介质的扰动理论来计算走时。首先简述各向异性介质的扰动理论，并推导非退化和退化情况的相速度扰动公式，而走时扰动可表示为沿未扰动介质中射线路径的积分；当未扰动介质选为均匀各向同性介质时，计算就较为简化。所给的数值例子证明了相速度扰动公式和走时计算公式的正确性。     

Mean Traveltime Curves Analysis: A Method to Improve Understanding of Data Behaviour in 2-D Transmission Tomography at the Pre-Inversion Stage

Transmission tomography methods show a great sensibility to data variability, which eventually includes data errors, often present in field experiments. Local optimization methods, traditionally used to solve this inverse problem, are very sensitive to these difficulties, failing to converge properly in the presence of spurious data. Regularization methods partially cope with these weaknesses, damping the instabilities.A complementary approach, adopted here, is to perform a structured analysis of data variability before the inversion, oriented to discriminate the contribution of errors from that of true geological heterogeneities. The key concept of mean traveltime curves ( and ) is introduced and described. Their analytical equations are deduced for isotropic homogeneous media and any recording geometry. Empirical mean traveltime curves can be inferred based solely on traveltime data, using the corresponding discrete estimators. The methodology proposed here proceeds through a user-defined subdivision of the domain of interest into isotropic homogeneous areas. Least squares velocity estimations and associated data misfits are used to scrutinize the behaviour of the implied source-receiver sets and of the ray-swept part of the geologic medium. Data are considered suspicious if zonal estimated velocities are non-consistent with a priori information. Also, independent fitting of both empirical curves helps to classify the genesis of the residuals: some situations are illustrated.Finally, we show the application of this technique to a data set from the Grimsel test site in Switzerland. Using this methodology, we detect some anomalous gathers, which may be responsible for the large range of velocities found in the initial imaging with this data set. Also, we give some indications of the possible sources of these anomalies. This approach offers a quick data variability analysis in the pre-processing stage, which, even if no data editing algorithms are finally used, always improves the understanding of the data structure.     

二维地层结构和速度的走时反演

介绍了在倾斜均匀各向同性层状介质情况下,根据地震反射走时反演二维地层结构和地层速度的地震射线层析成像方法——走时反演。正演模型射线追踪是根据Fermat原理实现的,即求解满足该原理的非线性方程组得到射线与界面的交点,进而计算相应的走时。反演是先假设一初始模型,用最优化方法使射线追踪走时与观察走时的残差极小。最后计算了分辨矩阵和信息密度矩阵,以评价反演结果。对有噪情形也进行了反演。     

渗透性水平向各向异性导致椭圆形地面沉降漏斗的一个性质

水平向渗透系数各向异性会导致抽水产生的水位降深等值线呈现椭圆形,进而产生椭圆形地面沉降漏斗,这种椭圆形地面沉降漏斗的现象在现实中广泛存在。渗透系数均质各向同性的圆形地面沉降漏斗中,地面某点沉降值s与该点至漏斗中心的距离r之间符合s- 线性关系,渗透性水平向各向异性导致的椭圆形地面沉降漏斗也具有s- 线性关系规律。应用地下水动力学相关理论,经过理论推导,得出了椭圆形地面沉降漏斗各径向线上半对数线性关系式 中常数项 、 与径向线方位角θ之间的函数关系式,即 、 与 或 的平方呈正比,该函数关系式得到中国、美国和印度共6个区域性地面沉降漏斗实例的有力验证。对公式验证回归所得的12个相关系数中,1个为87.19%,其余11个都在90%以上。以上渗透性水平向各向异性导致的区域性椭圆形地面沉降漏斗的性质,公式简单实用,可方便应用于椭圆形地面沉降漏斗中非监测点的沉降值推测,利于全面了解椭圆形沉降漏斗信息,具有广泛应用价值。     

VTI介质中的地层弹性参数反演

在VTI介质中，用非双曲线可以很好地近似P－P波中长排列情况下的旅行时；而对P－SV波来说，应用非双曲线却不能很好地近似P－SV波的施行时。对P－P波、P－SV波的旅行时进行三项泰勒展开时，它们的系数却包含了地震波的各向异性信息及其垂直速度信息，这为地震波的弹性参数反演提供了基础。实验分析表明，在中等排列长度下可以精确地得到了P－P波，P－SV波旅行时的三项泰勒展式系数，从而可以精确地进行地层弹性参数反演。     

纵波VTI介质各向异性参数的求取

在新疆塔里木盆地某地区的宽方位三维地震数据中,通过速度分析和不同炮检距范围数据的叠加剖面的比较,发现了各向同性基于双曲线时差曲线的NMO动校正时,远炮检距数据存在比较大的剩余动校正量。分析表明,是由于VTI介质火成岩的各向异性和垂向速度变化引起非双曲线时差。应用纵波短排列地震数据求取的均方根动校正速度,分别以扫描法和旅行时法得到了各自相互验证的各向异性参数。最后,利用所提取的各向异性参数,进行VTI介质各向异性速度分析、动校正和成像,剩余动校正得到消除,成像质量得到明显改善。     

基于快速匹配法的VTI介质走时计算

地震波全波形反演是当今地质构造反演的潮流。在层析成像等为其提供初始模型的预处理中,地震波初至走时是一个非常重要的物理量。因而,高效高精度且稳定的走时计算方法对于各向异性建模具有重要的研究意义。为实现高效高精度且稳定的走时计算,首先利用扰动理论及泰勒公式将具有垂直对称轴的横向各向同性（VTI）介质程函方程展开,得到走时解;然后引入各向同性快速推进法（fast marching method,FMM）,运用改进后的迎风差分格式求取差分格式黏滞解获取单点走时。结合窄带推进技术,得到了一种新的基于快速匹配法的VTI介质走时计算方法。通过对均匀弱各向异性模型计算结果和解析值的对比,评估了其误差,相对误差稳定于0.5%以下。针对该方法的有效性和稳定性,对层状介质模型和盐丘模型反射波走时进行了试算,取得了较好的效果。理论分析和模型试算表明,该方法对VTI介质走时计算具有较高的精度,能够应用于各向异性层析成像和全波场偏移等研究中。     

VTI介质多波速度与各向异性系数求取及应用

针对具有垂直对称轴的横向各向同性(VTI)介质,研究了多波速度与地层各向异性系数之间的关系。采用适应大偏移距以及强各向异性介质的双平方根方程,进行速度和各向异性系数分析,并将批量计算与交互解释相结合,实现了高精度各向异性介质多波速度与各向异性系数提取,并形成了一套利用多波地震资料求取地震波纵、横波速度及各向异性系数的方法和软件。所求取的参数可用于多波地震资料处理,提高多波资料的成像质量,也可为地层地质解释提供参考。实际资料的应用效果证明了该方法和软件的正确性及有效性。     

Ambient noise tomography with a large seismic array

The emergence of large-scale arrays of seismometers across several continents presents the opportunity to image the Earth's structure at unprecedented resolution, but methods must be developed to exploit the capabilities of these deployments. The capabilities and limitations of a method called “eikonal tomography” applied to ambient noise data are discussed here. In this method, surface wave wavefronts are tracked across an array and the gradient of the travel time field produces estimates of phase slowness and propagation direction. Application data from more than 1000 stations from EarthScope USArray in the central and western US and new Rayleigh wave isotropic and anisotropic phase velocity maps are presented together with an isotropic and azimuthally anisotropic 3D Vs model of the crust and uppermost mantle. As a ray theoretic method, eikonal tomography models bent rays but not other wavefield complexities. We present evidence, based on the systematics of an observed 1ψ component of anisotropy that we interpret as anisotropic bias caused by backscattering near an observing station, that finite frequency phenomena can be ignored in ambient noise tomography at periods shorter than ～ 40 to 50 s. At longer periods a higher order term based on wavefront amplitudes or finite frequency sensitivity kernels must be introduced if the amplitude of isotropic anomalies and the amplitude and fast-axis direction of azimuthal anisotropy are to be determined accurately.     

含倾斜定向排列裂隙介质纵波特征

根据Schoenberg的含裂隙介质的线性滑移模型在弱各向异性近似条件下根据Bond变换获得了含倾斜定向排列裂隙介质的弹性系数矩阵。在将含倾斜定向排列裂隙介质视为倾斜横向各项同性（TTI）介质的基础上根据弱各向异性近似条件下TTI介质纵波相速度、弹性波阻抗以及反射系数的近似解析解,研究了裂隙倾斜角对含倾斜定向排列裂隙介质中纵波相速度、反射系数以及波阻抗的影响。研究发现含倾斜裂隙介质的纵波相速度、各向异性弹性波阻抗、反射系数三个参数具有类似方位各向异性特征：入射角大的时候,显示较强方位各向异性,但是这三个参数对入射角的依赖程度却不完全相同,反射系数对入射角的依赖程度最大。     

各向异性介质相速度、群速度与克利斯托费尔方程

本文详细回顾了克利斯托费尔方程(Christoffel)的推导.在此基础上,提炼出时空域和频率域克利斯托费尔方程,前者可作正演计算介质的相速度,后者可计算介质的群速度.并计算了横观各向同性介质的相速度和群速度.最后对无损耗各向异性介质的相速度和群速度的关系进行了简单讨论.     

基于互相关函数对钻孔雷达层析成像的改进

常用的旅行时提取方法在针对大收发角度数据进行提取时,因其低信噪比而显得较为艰难且不够准确。提出一种新的基于互相关函数的方法对旅行时提取方法进行改进。首先对数据进行归一化并按角度对数据分类;然后对相同收发角度数据与该角度信噪比最大的数据进行互相关处理并叠加,从而得到该收发角度数据的参考波形;最后根据参考波形的初至波到达时间对该角度数据进行互相关处理,得到每道数据的初至波旅行时。通过计算机模拟的结果和对RAMAC雷达采集的实际数据进行分析处理的结果,对该方法的准确性进行了论证,与常规旅行时提取法得出的层析成像图相对比,该方法得到的层析成像图更为精确、清晰,能为判断地下异常提供更为有力的帮助。     

Analysis and comparison of mixed finite element and multipoint flux approximation methods for homogeneous media

We consider discretization on quadrilateral grids of an elliptic operator occurring, for example, in the pressure equation for porous-media flow. In a realistic setting – with non-orthogonal grid, and anisotropic, heterogeneous permeability – special discretization techniques are required. Mixed finite element (MFE) and multipoint flux approximation (MPFA) are two methods that can handle such situations. Previously, a framework for analytical comparison of MFE and MPFA in special cases has been suggested. A comparison of MFE and MPFA-O (one of two main variants of MPFA) for isotropic, homogeneous permeability on a uniformly distorted grid was also performed. In the current paper, we utilize the suggested framework in a slightly different manner to analyze and compare MFE, MPFA-O and MPFA-U (the second main variant of MPFA). We reconsider the case previously analyzed. We also consider the case of generally anisotropic, homogeneous permeability on an orthogonal grid.     

Angle-domain common-image gathers from anisotropic Gaussian beam migration and its application to anisotropy-induced imaging errors analysis

An approach for extracting angle-domain common-image gathers (ADCIGs) from anisotropic Gaussian beam prestack depth migration (GB-PSDM) is presented in this paper. The propagation angle is calculated in the process of migration using the real-value traveltime information of Gaussian beam. Based on the above, we further investigate the effects of anisotropy on GB-PSDM, where the corresponding ADCIGs are extracted to assess the quality of migration images. The test results of the VTI syncline model and the TTI thrust sheet model show that anisotropic parameters ε, δ, and tilt angle ??, have a great influence on the accuracy of the migrated image in anisotropic media, and ignoring any one of them will cause obvious imaging errors. The anisotropic GB-PSDM with the true anisotropic parameters can obtain more accurate seismic images of subsurface structures in anisotropic media.     

地震波走时计算的逆风差分算法

快速精确地旅行时计算在地震资料的叠前偏移与层析反演中起重要作用。利用逆足差分格式求解程函方程，首先在波阵面上寻找全局极小点，然后计算该点周围各点的地震波初至走时，同时，采用局部算法以确保计算的稳定性。该算法对于任意复杂地质模型都具有较高的精度。     

The effect of crustal anisotropy on reflector depth and velocity determination from wide-angle seismic data: a synthetic example based on South Island, New Zealand

When deriving velocity models by forward modelling or inverting travel time arrivals from seismic refraction data, a heterogeneous but isotropic earth is usually assumed. In regions where the earth is not isotropic at the scale at which it is being sampled, the assumption of isotropy can lead to significant errors in the velocities determined for the crust and the depths calculated to reflecting boundaries. Laboratory velocity measurements on rocks collected from the Haast Schist terrane of South Island, New Zealand, show significant (up to 20%) compressional (P) wave velocity anisotropy. Field data collected parallel and perpendicular to the foliation of the Haast Schist exhibit as much as 11% P-wave velocity anisotropy. We demonstrate, using finite-difference full-wavefield modelling, the types of errors and problems that might be encountered if isotropic methods are used to create velocity models from data collected in anisotropic regions. These reflector depth errors could be as much as 10–15% for a 10-km thick layer with significant (20%) P-wave velocity anisotropy. The implications for South Island, New Zealand, where the problem is compounded by extreme orientations of highly anisotropic rocks (foliation which varies from horizontal to near vertical), are considered. Finally, we discuss how the presence of a significant subsurface anisotropic body might manifest itself in wide-angle reflection/refraction and passive seismic datasets, and suggest ways in which such datasets may be used to determine the presence and extent of such anisotropic bodies.     

井间地震走时波形层析成像方法

提出了井间地震走时波形顺序反演方法。该方法先利用井间地震走时反演得到速度模型的低频成分 ,然后用井间地震波形反演获得速度模型的高频成分。数值模型试验和实际应用结果表明 ,该方法反演稳健 ,提高了走时成像的分辨率 ,克服了波形成像易于陷入局部极小的缺陷 ,实现了快速高分辨率成像。     

Geostatistical Analysis of Inverse Problem Variables: Application to Seismic Tomography

In this article we present a geostatistical approach to the transmission tomographic inverse problem, which is based on consideration of the inverse problem variables (velocity and traveltime errors) as regionalized variables (R.V.). Their structural analysis provides us with a new method to study the geophysical anisotropy of the rock, an important source of a priori information in order to design the anisotropic corrections. The underlying idea is that the geophysical structure can be deduced from the spatial structure of the regionalized variables which result from solving the tomographic problem with an isotropic algorithm. Also, the application of the structural analysis technique to the anisotropic corrected velocity field allows us to characterize the reliability of these corrections (model quality analysis). Geostatistical formalism also provides us with different techniques (parametric and non-parametric) to estimate and even simulate the velocity in the areas where this field has been considered anomalous based on field studies and on geophysical and statistical criteria. The kriging acts as a low-pass smoothing filter for the anomalous model parameters (velocities), but is not a substitute for an adequate filtering of the outliers before the inversion. This methodology opens the possibility of considering the inverse problem variables as stochastic processes, an important feature in cases where the tomogram is to be used as a tool of assessment to quantify the rock heterogeneities.     

利用线性旅行时插值射线追踪 计算近地表模型初至波走时

利用线性旅行时插值射线追踪对近地表模型进行正演计算,可以快捷、准确地获得初至波走时和射线路径。由于该算法计算的初至波不局限于折射波,因此很好地解决了浅层折射勘探中的低速"隐蔽层"问题;而且,由于该算法是基于网格划分和线性插值,因此它不仅可以追踪任意复杂介质的初至波,而且可以使得追踪的初至波射线路径逼于真实,避免了同类算法直接连接网络节点形成射线路径的缺陷(路径过于弯折,计算走时偏大)。将LTI算法同其他几种算法的追踪结果进行的对比和分析表明,LTI算法在计算初至波走时和射线路径方面较其他算法更为精准、稳定,是一种有效的射线追踪方法。