Grid travel-time tracing: Second-order method for the first arrivals in smooth media

A new computational scheme for calculating the first-arrival travel times on a rectangular grid of points is proposed. The new proposed method is of second-order accuracy. This means that the error of the calculated travel time is proportional to the second power of the grid spacing. The method should be sufficiently accurate for all applications in smooth seismic models. On the other hand, the method is not, in its present form, proposed for models with structural interfaces which make the method unstable and generate travel-time errors of the first order. Equations are also presented for the appropriate evaluation of the errors of calculated travel times to check their accuracy, and the proposed method is compared with other numerical methods. The method is developed, described and demonstrated in 2-D, but may also be extended to 3-D models and to general models with structural interfaces.     

Errors Due to the Common Ray Approximations of the Coupling Ray Theory

The common ray approximation considerably simplifies the numerical algorithm of the coupling ray theory for S waves, but may introduce errors in travel times due to the perturbation from the common reference ray. These travel-time errors can deteriorate the coupling-ray-theory solution at high frequencies. It is thus of principal importance for numerical applications to estimate the errors due to the common ray approximation.We derive the equations for estimating the travel-time errors due to the isotropic and anisotropic common ray approximations of the coupling ray theory. These equations represent the main result of the paper. The derivation is based on the general equations for the second-order perturbations of travel time. The accuracy of the anisotropic common ray approximation can be studied along the isotropic common rays, without tracing the anisotropic common rays.The derived equations are numerically tested in three 1-D models of differing degree of anisotropy. The first-order and second-order perturbation expansions of travel time from the isotropic common rays to anisotropic-ray-theory rays are compared with the anisotropic-ray-theory travel times. The errors due to the isotropic common ray approximation and due to the anisotropic common ray approximation are estimated. In the numerical example, the errors of the anisotropic common ray approximation are considerably smaller than the errors of the isotropic common ray approximation.The effect of the isotropic common ray approximation on the coupling-ray-theory synthetic seismograms is demonstrated graphically. For comparison, the effects of the quasi-isotropic projection of the Green tensor, of the quasi-isotropic approximation of the Christoffel matrix, and of the quasi-isotropic perturbation of travel times on the coupling-ray-theory synthetic seismograms are also shown. The projection of the travel-time errors on the relative errors of the time-harmonic Green tensor is briefly presented.     

Finite-difference solution of the transport equation: First results

The ray-theoretical transport equation for inhomogeneous isotropic media (2D-SH case) is solved by the method of finite differences on a rectangular grid, both for an incident plane wave (explicit scheme) and a line source (implicit scheme). Results for homogeneous models and for heterogeneous models with structural discontinuities are discussed. First-arrival travel times calculated by various techniques serve as input for the solution of the transport equation and the computation of amplitudes of first arrivals. To obtain correct amplitudes the travel times must be highly accurate and the discontinuities must be smoothed out; the reason is that the spatial second derivatives of the travel time field enter the transport equation. In the simple cases studied, finite differences provide a fast and efficient tool for the computation of first-arrival amplitudes.     

True Amplitude Migration Weights from Travel Times

— 3-D amplitude preserving prestack migration of the Kirchhoff type is a task of high computational effort. A substantial part of this effort is spent on the calculation of proper weight functions for the diffraction stack. We propose a new strategy to compute the migration weights directly from coarse gridded travel-time data which are in any event needed for the summation along diffraction time surfaces. The technique employs second-order travel-time derivatives that contain all necessary information on the weight functions. Their determination alone from travel times significantly reduces the requirements in computational time and particularly storage, since it is done on the fly. Application of the method shows good accordance between numerical and analytical results for the simple types of models considered in this study.     

Transformation relations for second-order derivatives of travel time in anisotropic media

In the computation of paraxial travel times and Gaussian beams, the basic role is played by the second-order derivatives of the travel-time field at the reference ray. These derivatives can be determined by dynamic ray tracing (DRT) along the ray. Two basic DRT systems have been broadly used in applications: the DRT system in Cartesian coordinates and the DRT system in ray-centred coordinates. In this paper, the transformation relations between the second-order derivatives of the travel-time field in Cartesian and ray-centred coordinates are derived. These transformation relations can be used both in isotropic and anisotropic media, including computations of complex-valued travel times necessary for the evaluation of Gaussian beams.     

用sPn震相测定近震震源深度

用近震深度震相ｓＰｎ与Ｐｎ波的到时差测定近震深度，方法以求简捷准确。为此，对我国部分地区台网记录的ｓＰｎ震相进行了初步分析研究，通过研制的计算ｓＰｎ走时程序运算，给出华北、山西、华东、西北及四川地区的ｓＰｎ－Ｐｎ走时差对应震源深（ｈ）表和ｓＰｎ－Ｐｎ求震源深度列线图。经过震例检验，该表比较适合本地区的走时特性。为应用ｓＰｎ震相测准我国部分地区震源深度，提供了有利工具。     

Analysis of Observed and Predicted Tsunami Travel Times for the Pacific and Indian Oceans

I have examined over 1500 historical tsunami travel-time records for 127 tsunamigenic earthquakes that occurred in the Pacific and Indian Oceans. After subjecting the observations to simple tests to rule out gross errors I compare the remaining reports to simple travel-time predictions using Huygens method and the long-wave approximation, thus simulating the calculations that typically take place in a tsunami warning situation. In general, I find a high correspondence between predicted and reported travel times however, significant departures exist. Some outliers imply significantly slower propagation speeds than predicted; many of these are clearly the consequences of observers not being able to detect the (possibly weak?) first arrivals. Other outliers imply excessively long predicted travel times. These outliers reflect peculiar geometric and bathymetric conditions that are poorly represented in global bathymetric grids, leading to longer propagation paths and consequently increased travel times. Analysis of Δt, the difference between observed and predicted travel time, yields a mean Δt of 19 minutes with a standard deviation of 131 minutes. Robust statistics, being less sensitive to outliers, yield a median Δt of just 18 seconds and a median absolute deviation of 33 minutes. Care is needed to process bathymetry to avoid excessive travel-time delays in shallow areas. I also show that a 2×2 arc minute grid yields better results that a 5×5 arc minute grid; the latter in general yielding slightly slower propagation predictions. The largest remaining source of error appears to be the inadequacy of the point-source approximation to the finite tsunami-generating area.     

Regional travel-time and slowness measurements for P-waves in the distance range 40–57°

A method for analysing travel times measured at a large array or a network of seismographs from many earthquakes within a specific region has been developed. Approximate relative station corrections are calculated from the residuals on a least-squares line or least-squares quadratic form fitted through the times for each earthquake, and may be improved by iteration after a preliminary travel-time curve has been derived. Accurate relative baseline corrections for each earthquake are also calculated iteratively, and an optimum slowness-distance curve is determined from the combined corrected travel times from all earthquakes using a trade-off procedure. Calculations using synthetic travel-time data suggest that abrupt changes in slowness of ～ 0.4 s deg^?1 due to the presence of triplications are generally resolvable, provided that the effects of lateral variations are small, even with random epicentre mislocations in the range ± 0.5°. Slowness measurements at a network of temporary stations deployed across Australia do not show any discontinuities in slowness greater than 0.2 s deg^?1 in the distance range 45–54°. Similar measurements at the Warramunga array from the same source regions, however, suggest the presence of complexity in the slowness curve at distances close to 50°. Relative arrival times at the temporary network generally have standard deviations less than 0.25 s, thus suggesting that details of structure finer than those derived from conventional travel-time studies can be resolved.     

Earthquake location in transversely isotropic media with a tilted symmetry axis

The conventional intersection method for earthquake location in isotropic media is developed in the case of transversely isotropic media with a tilted symmetry axis (TTI media). The hypocenter is determined using its loci, which are calculated through a minimum travel time tree algorithm for ray tracing in TTI media. There are no restrictions on the structural complexity of the model or on the anisotropy strength of the medium. The location method is validated by its application to determine the hypocenter and origin time of an event in a complex TTI structure, in accordance with four hypotheses or study cases: (a) accurate model and arrival times, (b) perturbed model with randomly variable elastic parameter, (c) noisy arrival time data, and (d) incomplete set of observations from the seismic stations. Furthermore, several numerical tests demonstrate that the orientation of the symmetry axis has a significant effect on the hypocenter location when the seismic anisotropy is not very weak. Moreover, if the hypocentral determination is based on an isotropic reference model while the real medium is anisotropic, the resultant location errors can be considerable even though the anisotropy strength does not exceed 6.10%.     

P-wave Tomography in Inhomogeneous Orthorhombic Media

— A P-wave tomographic method for 3-D complex media (3-D distribution of elastic parameters and curved interfaces) with orthorhombic symmetry is presented in this paper. The technique uses an iterative linear approach to the nonlinear travel-time inversion problem. The hypothesis of orthorhombic anisotropy and 3-D inhomogeneity increases the set of parameters describing the model dramatically compared to the isotropic case. Assuming a Factorized Anisotropic Inhomogeneous (FAI) medium and weak anisotropy, we solve the forward problem by a perturbation approach. We use a finite element approach in which the FAI medium is divided into a set of elements with polynomial elastic parameter distributions. Inside each element, analytical expressions for rays and travel times, valid to first-order, are given for P waves in orthorhombic inhomogeneous media. More complex media can be modeled by introducing interfaces separating FAI media with different elastic properties. Simple formulae are given for the Fréchet derivatives of the travel time with respect to the elastic parameters and the interface parameters. In the weak anisotropy hypothesis the P-wave travel times are sensitive only to a subset of the orthorhombic parameters: the six P-wave elastic parameters and the three Euler angles defining the orientation of the mirror planes of symmetry. The P-wave travel times are inverted by minimizing in terms of least-squares the misfit between the observed and calculated travel times. The solution is approached using a Singular Value Decomposition (SVD). The stability of the inversion is ensured by making use of suitable a priori information and/or by applying regularization. The technique is applied to two synthetic data sets, simulating simple Vertical Seismic Profile (VSP) experiments. The examples demonstrate the necessity of good 3-D ray coverage when considering complex anisotropic symmetry.     

METHOD FOR DETERMINATION OF VELOCITY AND DEPTH FROM SEISMIC REFLECTION DATA1

The estimation of velocity and depth is an important stage in seismic data processing and interpretation. We present a method for velocity-depth model estimation from unstacked data. This method is formulated as an iterative algorithm producing a model which maximizes some measure of coherency computed along traveltimes generated by tracing rays through the model. In the model the interfaces are represented as cubic splines and it is assumed that the velocity in each layer is constant. The inversion includes the determination of the velocities in all the layers and the location of the spline knots. The process input consists of unstacked seismic data and an initial velocity-depth model. This model is often based on nearby well information and an interpretation of the stacked section. Inversion is performed iteratively layer after layer; during each iteration synthetic travel-time curves are calculated for the interface under consideration. A functional characterizing the main correlation properties of the wavefield is then formed along the synthetic arrival times. It is assumed that the functional reaches a maximum value when the synthetic arrival time curves match the arrival times of the events on the field gathers. The maximum value of the functional is obtained by an effective algorithm of non-linear programming. The present inversion algorithm has the advantages that event picking on the unstacked data is not required and is not based on curve fitting of hyperbolic approximations of the arrival times. The method has been successfully applied to both synthetic and field data.     

Errors due to the anisotropic-common-ray approximation of the coupling ray theory

The common-ray approximation eliminates problems with ray tracing through S-wave singularities and also considerably simplifies the numerical algorithm of the coupling ray theory for S waves, but may introduce errors in travel times due to the perturbation from the common reference ray. These travel-time errors can deteriorate the coupling-ray-theory solution at high frequencies. It is thus of principal importance for numerical applications to estimate the errors due to the common-ray approximation applied. The anisotropic-common-ray approximation of the coupling ray theory is more accurate than the isotropic-common-ray approximation. We derive the equations for estimating the travel-time errors due to the anisotropic-common-ray (and also isotropic-common-ray) approximation of the coupling ray theory. The errors of the common-ray approximations are calculated along the anisotropic common rays in smooth velocity models without interfaces. The derivation is based on the general equations for the second-order perturbations of travel time.     

数字化地震记录震相自动识别的方法研究

针对目前震相自动识别方法不能自动给出震相识别区间,以及不能确定识别出震相名称的问题,运用震相的运动学特征,由平均速度模型和J-B走时表数据,自动计算近震、远震和极远震的震相走时及震中距。对多尺度小波分解进行单支重构作为识别不同震相的分析信号。先求出初至震相和最大面波到时,估算出震中距,然后找到S波或PP波到时,求出准确的震中距,即可自动给出各震相的识别区间,采用线性偏振法在给定区间中识别出震相的初至时刻。由于该方法采用的是先明确要找什么震相,再由该震相的走时确定寻找区间,所以找出的初至就是要寻找的震相,自然解决了识别出震相的名称问题,从而实现了对震相的全程自动识别。     

基于射线追踪技术计算地震定位中震源轨迹的改进方法

使用震源轨迹确定震源位置不仅稳健而且直观,但当介质复杂时震源轨迹难以给出解析解.基于最小走时树射线追踪技术计算震源轨迹的方法(以轨迹所在的残差场中残差最小的点(初始点)至残差较小的点(震源轨迹代表点)的射线路径表示震源轨迹)适用于复杂速度模型,但尚不能正确计算由多段组成的震源轨迹,同时兼顾计算轨迹的完整性和精细性较为困难,计算参数设置烦琐不适于大批量数据的自动处理.针对该方法存在的问题,本文对其进行了改进:(1)采用一种"削皮"算法选取震源轨迹所经过的模型单元的节点作为轨迹代表点;(2)将残差较小的区域作为震源轨迹计算区域(该区域依轨迹分布自适应地划分为若干个连通区域),从未计算的轨迹代表点中选取残差最小者作为射线路径初始点,利用最小走时树算法依次计算所有连通区域内的震源轨迹;(3)通过去掉较短的不再分叉的射线路径使震源轨迹更为精细.虚拟和真实事件的算例表明,改进方法有效克服了原方法的不足,可便捷地计算复杂速度模型中事件的震源轨迹,计算的轨迹精细且较完整.     

Seismic phase picking in China Seismic Array using a deep convolutional neuron network*

Seismic phase picking is the preliminary work of earthquake location and body-wave travel time tomography. Manual picking is considered as the most accurate way to access the arrival times but time consuming. Many automatic picking methods were proposed in the past decades, but their precisions are not as high as human experts especially for events with low ratio of signal to noise and later arrivals. As the increasing deployment of large seismic array, the existing methods can not meet the requirements of quick and accurate phase picking. In this study, we applied a phase picking algorithm developed on the base of deep convolutional neuron network (PickNet) to pick seismic phase arrivals in ChinArray-Phase III. The comparison of picking error of PickNet and the traditional method shows that PickNet is capable of picking more precise phases and can be applied in a large dense array. The raw picked travel-time data shows a large variation deviated from the traveltime curves. The absolute location residual is a key criteria for travel-time data selection. Besides, we proposed a flowchart to determine the accurate location of the single-station earthquake via dense seismic array and phase arrival picked by PickNet. This research expands the phase arrival dataset and improves the location accuracy of single-station earthquake.     

COMMON OFFSET PLANE MIGRATION (COPMIG)*

The quality of results of migration before stack is sensitive to inaccuracies in the velocity field applied. This does not hold if only traces of similar sources-receiver distances (common offset traces) enter the migration process. In this case, velocity deviations generate minor shifts in travel times of migrated interfaces but no deterioration in quality. These time shifts are proportional to both the velocity error and the square of the source-receiver distance. The above observations suggest the following migration scheme: migrate separately the traces of the various common offset planes or groups of neighbouring common offset planes; for every common midpoint plane and as a function of travel-time perform a residual NMO search to find trajectories t) =t)_o+px)² of maximum coherency along which migrated events are aligned; correct for residual NMO and stack the migration results obtained in the various common offset planes to obtain the final migration result. This process not only takes care of inaccurate migration velocities but also corrects partly for effects of refraction. It is shown by means of an example that good migration results are generated even with a considerably deviating velocity field.     

A note on two-point paraxial travel times

Recently, several expressions for the two-point paraxial travel time in laterally varying, isotropic or anisotropic layered media were derived. The two-point paraxial travel time gives the travel time from point S′ to point R′, both these points being situated close to a known reference ray Ω, along which the ray-propagator matrix was calculated by dynamic ray tracing. The reference ray and the position of points S′ and R′ are specified in Cartesian coordinates. Two such expressions for the two-point paraxial travel time play an important role. The first is based on the 4 × 4 ray propagator matrix, computed by dynamic ray tracing along the reference ray in ray-centred coordinates. The second requires the knowledge of the 6 × 6 ray propagator matrix computed by dynamic ray tracing along the reference ray in Cartesian coordinates. Both expressions were derived fully independently, using different methods, and are expressed in quite different forms. In this paper we prove that the two expressions are fully equivalent and can be transformed into each other.     

中国地震台网初至P波区域三维走时表的建立

基于LLNL-G3Dv3全球P波三维速度模型,应用FMM软件包计算并建立了中国地震台网990个台站的初至P波区域三维走时表．该走时表覆盖了以台站为中心的水平向20°×20°、 垂直向-5.1—80 km （向下为正）的三维空间. 其水平向间隔为0.2°,垂直向间隔为5 km．这样对于任一深度小于80 km的震源,均可以应用此三维走时表计算其到周围10°范围内台站的走时．中国地震台网初至P波区域三维走时表的建立,对于改善区域初至P波走时预测,提高地震定位精度有一定现实意义．      

宽角反射地震波走时模拟的双重网格法

在研究地壳结构的人工源宽角反射地震资料解释中,常规宽角反射波走时和射线路径计算大都假定地壳模型为层状块状均匀介质.为了逼近实际地壳结构模型,要求模型尺度较大,为了提高地震资料解释的可靠性,须减小模型离散单元的尺寸,但同时计算量大大增加,使资料解释的效率较低.为此,本文尝试同时提高宽角反射地震资料解释效率和可靠性的方法,即使用双重网格计算宽角反射地震波走时和射线路径的最小走时树方法.双重网格法在均匀介质内部仅计算大网格节点,在速度变化点、震源点和检波点区域,同时计算小网格节点;在界面边界点使用比介质内部节点更大的子波传播区域.模型计算结果表明,对于大尺度的层状块状均匀介质模型,在保证精度的条件下,本文所提出的双重网格射线追踪方法的计算效率比单网格方法显著提高.     

基于LTI和网格界面剖分的三维地震射线追踪算法

将二维线性走时插值射线追踪算法(LTI)推广应用至三维模型,并结合网格界面剖分方式,提出了一种三维射线追踪算法．该算法既可获得高精度的全局最小走时和射线路径,又具有快速稳定的特点．三维模型计算结果表明,在模型参数包括网格密度完全相同情况下,本文算法较传统的三维最短路径算法在计算效率、走时和射线的计算精度上均有明显改进．