A note on the calculation of travel times in a transversely isotropic Earth model

Some explicit formulae are presented for the calculation of travel times in transversely isotropic Earth models.     

一种分析岩土模型地震受力变形的方法

在国内外岩土模型动力试验中,一般以图表形式对监测区域进行逐一动态分析,无法直观形象的掌握试验模型整体受力变形,为深入分析动态响应机理带来一定的不便。以前-后排抗滑桩加固斜坡桥基的大型振动台模型试验为例,通过在岩土体中埋设一定数量的自制磷青铜带和水平加速度计,监测滑坡变形和加速度响应,结合已知测点的水平坐标和竖向坐标,利用Renka Cline随机矩阵生成方法转换为数字矩阵形式,据此绘制坡体变形及PGA放大系数的二维等势图。试验结果显示,二维等势图能合理反映斜坡PGA放大系数的变化规律,揭示振动波作用下斜坡变形破坏的基本特征,研究结论和试验现象保持一致,满足斜坡模型整体受力变形分析的基本要求,可以作为一种实用的试验分析方法。     

Interpretation of first arrival travel times in seismic refraction work

The necessary condition for the seismic refraction method to succeed is that the refracted first arrivals from each layer in a multilayered earth system should be detected on a seismogram as first arrivals, and this is possible only when velocities of all underlying layers are successively greater. The usual procedure to interpret the refraction travel times is to fit such a data set with several intersecting straight lines by employing a visual technique which may lead to errors of subjective judgment, as the velocity model depends on the selection of various line segments through the data. To remove the visual fit we propose here a layer stripping method based on minimum intercept time, apparent velocity, rms residual, and maximum data points by least-squares fitting to yield several intersecting straight lines. Once data are segmented out, the conventional equations can be used to determine the velocity structure.     

基于克里金方法的区域震相走时标定技术研究div.

在台网比较稀疏的情况下,台站走时标定是提高低震级事件定位、识别能力的重要手段．为了提高稀疏台网的定位精度,首先利用标定事件和IASPEI1991走时表以及中国大陆走时表,计算了初至P波和Lg波到AAK,MAKZ,NIL,TLG以及WMQ等5个台站的走时残差;接着,采用非平稳贝叶斯克里金方法和走时残差数据构造上述台站的走时校正曲面;最后,通过加入和不加入走时校正定位一系列位置准确的发生在新疆地区的地震事件评估克里金走时校正的有效性．结果表明,克里金走时校正能够较大地提高稀疏台网的定位精度,同时有效地缩小误差椭圆的面积．      

A test of new Earth models against PcP and PmKP travel times

PcP and PmKP travel times are computed for three simple or parametric Earth models, based on free-oscillation and travel-time data B₁, PEM-A and HB₁ and compared with PcP and PmKP travel times from different sources. This comparison is made only for the region above and below the core-mantle boundary and is of interest because of the current search for a standard Earth model. The comparison shows that only model B₁ does not need a correction for its PcP travel times. For the PmKP travel times for the three models, corrections of the form Δt = a + bm were obtained. The models need the following corrections for b: ?1.3 for B₁, 2.8 for HB₁ and 0.6 for PEM-A. The corrections a are shown to be equal to the observed corrections for PcP at large epicentral distances. The inversions of free-oscillation data to obtain Earth models are most successful when body-wave phases that interact with the core are included.     

Combined inversion of first-arrival travel times and reflection travel times

In this work, we propose a method for determining reflection travel times based on the acquisition of first-arrival travel times via the fast sweeping method. The accuracy of this scheme was proven by conducting model experiments to establish a foundation for first-arrival tomography, reflection tomography and combined tomography. Reflection tomography was subsequently achieved using the adjoint-state method; on this basis, we propose a combined tomography method involving both first-arrival and reflection tomography. In the model experiments, excellent results were obtained via first-arrival tomography, reflection tomography and our combined tomography method. Finally, full-waveform inversion was performed, with the inversion produced by combined tomography used as the initial model. Excellent results were obtained using this approach. However, combined tomography reproduced and characterized the model much more realistically.     

A hybrid approach to compute seismic travel times in three-dimensional tetrahedral meshes

We propose an optimized method to compute travel times for seismic inversion problems. It is a hybrid method combining several approaches to deal with travel time computation accuracy in unstructured meshes based on tetrahedral elementary cells. As in the linear travel time interpolation method, the proposed approach computes travel times using seismic ray paths. The method operates in two sequential steps: At a first stage, travel times are computed for all nodes of the mesh using a modified version of the shortest path method. The difference with the standard version is that additional secondary nodes (called tertiary nodes) are added temporarily around seismic sources in order to improve accuracy with a reasonable increase in computational cost. During the second step, the steepest travel time gradient method is used to trace back ray paths for each source–receiver pair. Travel times at each receiver are then recomputed using slowness values at the intersection points between the ray path and the traversed cells. A number of numerical tests with an array of different velocity models, mesh resolutions and mesh topologies have been carried out. These tests showed that an average relative error in the order of 0.1% can be achieved at a computational cost that is suitable for travel time inversion.     

An empirical method for estimating travel times for wet volcanic mass flows

 Travel times for wet volcanic mass flows (debris avalanches and lahars) can be forecast as a function of distance from source when the approximate flow rate (peak discharge near the source) can be estimated beforehand. The near-source flow rate is primarily a function of initial flow volume, which should be possible to estimate to an order of magnitude on the basis of geologic, geomorphic, and hydrologic factors at a particular volcano. Least-squares best fits to plots of flow-front travel time as a function of distance from source provide predictive second-degree polynomial equations with high coefficients of determination for four broad size classes of flow based on near-source flow rate: extremely large flows (>1 000 000 m³/s), very large flows (10 000–1 000 000 m³/s), large flows (1000–10 000 m³/s), and moderate flows (100–1000 m³/s). A strong nonlinear correlation that exists between initial total flow volume and flow rate for "instantaneously" generated debris flows can be used to estimate near-source flow rates in advance. Differences in geomorphic controlling factors among different flows in the data sets have relatively little effect on the strong nonlinear correlations between travel time and distance from source. Differences in flow type may be important, especially for extremely large flows, but this could not be evaluated here. At a given distance away from a volcano, travel times can vary by approximately an order of magnitude depending on flow rate. The method can provide emergency-management officials a means for estimating time windows for evacuation of communities located in hazard zones downstream from potentially hazardous volcanoes. Received: 5 June 1997 / Accepted: 2 February 1998     

An upwind fast sweeping scheme for calculating seismic wave first‐arrival travel times for models with an irregular free surface

The topography‐dependent eikonal equation formulated in a curvilinear coordinate system has recently been established and revealed as being effective in calculating first‐arrival travel times of seismic waves in an Earth model with an irregular free surface. The Lax–Friedrichs sweeping scheme, widely used in previous studies as for approximating the topography‐dependent eikonal equation viscosity solutions, is more dissipative and needs a much higher number of iterations to converge. Furthermore, the required number of iterations grows with the grid refinement and results in heavy computation in dense grids, which hampers the application of the Lax–Friedrichs sweeping scheme to seismic wave travel‐time calculation and high‐resolution imaging. In this paper, we introduce a new upwind fast sweeping solver by discretising the Legendre transform of the numerical Hamiltonian of the topography‐dependent eikonal equation using an explicit formula. The minimisation related to the Legendre transform in the sweeping scheme is solved analytically, which proved to be much more efficient than the Lax–Friedrichs algorithm in solving the topography‐dependent eikonal equation. Several numerical experiments demonstrate that the new upwind fast sweeping method converges and achieves much better accuracy after a finite number of iterations, independently of the mesh size, which makes it an efficient and robust tool for calculating travel times in the presence of a non‐flat free surface.     

An analytical model for ice-edge upwelling

Abstract

This paper presents an analytical, two-dimensional model of the wind-induced homogeneous circulation near the edge of an ice pack floating on the ocean surface. It is shown that a vertical shear layer arises under the ice edge, by which the wind-driven geostrophic motion in the open ocean is matched to the flow region underneath the ice. As in coastal upwelling models, this shear layer consists of a thin E ^1/2-layer inside a thicker E ^1/4-layer (E being the Ekman number). Under certain conditions the shear layer produces a vertical mass flux from the bottom to the surface Ekman layer. Near the surface this upwelling flux is concentrated in the narrow E ^1/2-layer. Comparison with observations of upwelling at the edge of a polar ice pack shows good agreement.     

Joint inversion of travel times and waveforms by means of annealing algorithms. Application to a seismic refraction experiment

An algorithm of annealing is applied to a joint inversion of travel times and waveforms belonging to a synthetically generated seismic refraction experiment. The medium (crust and upper mantle) is modelled by a set of plane stratified layers and a halfspace. The obtained structure (elastic parameters and depth of layers) shows that, in spite of contaminating seismic noise and poor knowledge of the seismic source, annealing methods are a good tool in these kinds of inversion problems. We think that many characteristics of the annealing process described here could be used with real data and more sophisticated media for the crust and upper mantle of the earth than the present example.     

自重可压缩层状球体地球模型下地震变形算法的优化研究

本文利用龙格库塔法求解球坐标系下微分方程组, 基于自重、可压缩和连续分层地球模型研究地震引起的地球变形, 优化相关算法.首先对传统的两次同向积分法进行调整, 对微分方程的齐次部分进行两次相向积分, 分别由地心向上积分至震源和由地表向下积分至震源.接着利用震源间断条件求解震源上方和震源下方通解的组合系数, 得到微分方程组的解矢量.基于该积分方法可以避免高阶微分方程组求解中精度丢失的问题.接着与均质地球模型下微分方程组解析解的比较, 发现自重效应对走滑震源低阶求解结果的影响较大, 对垂直引张震源变形解y₄的影响超过100阶.最后比较了均质地球模型和层状地球模型下高阶求解结果, 发现两者的比值趋于稳定, 当均质地球模型参数设置为震源所在层参数时, 并且变形深度和震源处于模型同一层, 两种地球模型下高阶变形解的比值趋于1.基于该发现, 结合均质地球模型下渐近解, 可解决实际分层地球模型下震源附近地震变形发散的问题.

     

Interpretation of seismic reflection records: Direct calculation of interval velocities and layer thicknesses from travel times

In reflection surveys and velocity analysis, calculations of interval velocities and layer-thicknesses of a multilayered horizontal structure are often based on Dix's equation which requires the travel times at zero offsets and a prior estimate of the root mean squared velocities.In this paper a method is presented which requires only the reflection travel-time data. A set of equations are derived which relate the interval velocity and thickness of a layer to the reflection travel time from the top and the bottom of that layer, the offset distances and the ray parameter. It is shown that the difference of the offset distances and the difference of the picked travel times of any reflected rays with the same value of ray parameter from the top and the bottom of a horizontal layer can be used to calculate the interval velocity and thickness of that layer.     

An analytical method for the evaluation of the in-plan irregularity of non-regularly asymmetric buildings

This paper describes a new method for the evaluation of the static eccentricity $e_{s}$ and the ratio $\Omega _{\uptheta } $ of uncoupled torsional to lateral frequencies in real multi-storey buildings. The above-mentioned parameters greatly affect the lateral-to-torsional coupling of the response of asymmetric systems and thus are of paramount importance in the assessment of the in-plan irregularity of buildings. The proposed method, which is a generalization of that suggested by Calderoni et al. (Earthq Spectra 18(2):219–231, 2002), allows the calculation of the static eccentricity $e_{s}$ and the ratio $\Omega _{\uptheta } $ from the structural response to arbitrary distributions of forces and torsional couples. The effectiveness of the method is validated on some regularly and non-regularly asymmetric buildings characterised by different in-plan irregularity. The analyses demonstrate that the results of the method are rigorous in the case of regularly asymmetric systems and only slightly depend upon the heightwise distribution of the forces in the case of non-regularly asymmetric systems. Finally, the values of the static eccentricity $e_{s}$ and the ratio $\Omega _{\uptheta } $ resulting from the proposed method are compared to those obtained by means of the procedure suggested by Makarios and Anastassiadis in (Struct Des Tall Spec Build 7(1):33–55, 1998a; Struct Des Tall Spec Build 7(1):57–71, 1998b) .     

An ordered probit model for seismic intensity data

Seismic intensity, measured through the Mercalli–Cancani–Sieberg (MCS) scale, provides an assessment of ground shaking level deduced from building damages, any natural environment changes and from any observed effects or feelings. Generally, moving away from the earthquake epicentre, the effects are lower but intensities may vary in space, as there could be areas that amplify or reduce the shaking depending on the earthquake source geometry, geological features and local factors. Currently, the Istituto Nazionale di Geofisica e Vulcanologia analyzes, for each seismic event, intensity data collected through the online macroseismic questionnaire available at the web-page www.haisentitoilterremoto.it. Questionnaire responses are aggregated at the municipality level and analyzed to obtain an intensity defined on an ordinal categorical scale. The main aim of this work is to model macroseismic attenuation and obtain an intensity prediction equation which describes the decay of macroseismic intensity as a function of the magnitude and distance from the hypocentre. To do this we employ an ordered probit model, assuming that the intensity response variable is related through the link probit function to some predictors. Differently from what it is commonly done in the macroseismic literature, this approach takes properly into account the qualitative and ordinal nature of the macroseismic intensity as defined on the MCS scale. Using Markov chain Monte Carlo methods, we estimate the posterior probability of the intensity at each site. Moreover, by comparing observed and estimated intensities we are able to detect anomalous areas in terms of residuals. This kind of information can be useful for a better assessment of seismic risk and for promoting effective policies to reduce major damages.     

Constraining the anisotropy structure of the crust by joint inversion of seismic reflection travel times and wave polarizations

In the context of wide-angle seismic profiling, the determination of the physical properties of the Earth crust, such as the elastic layer depth and seismic velocity, is often performed by inversion of P- and/or S-phases propagation data supplying the geometry of the medium (reflector depths) or any other structural parameter (P- or S-wave velocity, density...). Moreover, the inversion for velocity structure and interfaces is commonly performed using only seismic reflection travel times and/or crustal phase amplitudes in isotropic media. But it is very important to utilize more available information to constrain the non-uniqueness of the solution. In this paper, we present a simultaneous inversion method of seismic reflection travel times and polarizations data of transient elastic waves in stratified media to reconstruct not only layer depth and vertical P-wave velocity but also the anisotropy feature of the crust based on the estimation of the Thomsen’s parameters. We carry out a checking with synthetic data, comparing the inversion results obtained by anisotropic travel-time inversion to the results derived by joint inversion of seismic reflection travel times and polarizations data. The comparison proves that the first procedure leads to biased anisotropic models, while the second one fits nearly the real model. This makes the joint inversion method feasible. Finally, we investigate the geometry, P-wave velocity structure and anisotropy of the crust beneath Southeastern China by applying the proposed inversion method to previously acquired wide-angle seismic data. In this case, the anisotropy signature provides clear evidence that the Jiangshan-Shaoxing fault is the natural boundary between the Yangtze and Cathaysia blocks.     

An advanced analytical model for high damping rubber bearings

Base‐isolation technologies have been developed over the years in attempts to mitigate the effects of earthquakes on structures and potentially vulnerable contents in earthquake prone areas of the world. The high damping rubber bearing (HDRB) is a relatively recent and evolving technology of this kind. The isolator shifts the fundamental period of the base‐isolated structure to a value beyond the range of the plentiful energy‐containing periods of earthquake motions and supplies significant damping to dissipate energy caused by motions. Nevertheless, the highly non‐linear mechanical behaviour of the HDRB is so complex, especially at large strains, that it is difficult to model it analytically. In this paper, an extensive study of experimental tests for identifying the mechanical characteristics of the HDRB is presented. By modifying the Wen's model to include the rate‐dependent effects, an advanced analytical model in an incremental form for the HDRB is also proposed. A very good agreement between the analytical and experimental results has been obtained. It is illustrated that the proposed mathematical model can predict well the mechanical behaviour of HDRB bearings, even at large shear strain. Copyright © 2003 John Wiley & Sons, Ltd.     

Inversion of travel times obtained during active seismic refraction experiments CELEBRATION 2000, ALP 2002 and SUDETES 2003

A series of kinematic inversions based on robust non-linear optimization approach were performed using travel time data from a series of seismic refraction experiments: CELEBRATION 2000, ALP 2002 and SUDETES 2003. These experiments were performed in Central Europe from 2000 to 2003. Data from 8 profiles (CEL09, CEL10, Alp01, S01, S02, S03, S04 and S05) were processed in this study. The goal of this work was to find seismic velocity models yielding travel times consistent with observed data. Optimum 2D inhomogeneous isotropic P-wave velocity models were computed. We have developed and used a specialized two-step inverse procedure. In the first “parametric” step, the velocity model contains interfaces whose shapes are defined by a number of parameters. The velocity along each interface is supposed to be constant but may be different along the upper and lower side of the interface. Linear vertical interpolation is used for points in between interfaces. All parameters are searched for using robust non-linear optimization (Differential Evolution algorithm). Rays are continuously traced by the bending technique. In the second “tomographic” step, small-scale velocity perturbations are introduced in a dense grid covering the currently obtained velocity model. Rays are fixed in this step. Final velocity models yield travel time residuals comparable to typical picking errors (RMS ∼ 0.1 s). As a result, depth-velocity cross-sections of P waves along all processed profiles are obtained. The depth range of the models is 35–50 km, the velocity varies in the range 3.5–8.2 km/s. Lowest velocities are detected in near-surface depth sections crossing sedimentary formations. The middle crust is generally more homogeneous and has typical P wave velocity around 6 km/s. Surprisingly the lower crust is less homogeneous and the computed velocity is in the range 6.5–7.5 km/s. The MOHO is detected in the depth ≈30–45 km.     

Synthetic seismograms for finite sources in spherically symmetric Earth using normal-mode summation

Normal-mode summation is the most rapidly used method in calculating synthetic seismograms. However, normal-mode summation is mostly applied to point sources. For earthquakes triggered by faults extending for as long as several 100 km, the seismic waves are usually simulated by point source summation. In this paper, we attempt to follow a different route, i.e., directly calculate the excitation of each mode, and use normal-mode summation to obtain the seismogram. Furthermore, we assume the finite source to be a ‘‘line source' and numerically calculate the transverse component of synthetic seismograms for vertical strike-slip faults. Finally, we analyze the features in the Love waves excited by finite faults.     

基于交叉梯度交替结构约束的二维地震走时与全通道直流电阻率联合反演

在利用不同的地球物理勘探方法对地下复杂介质成像时,因观测系统的非完备性及数据本身对某些岩石物性的不敏感性,单独成像的结果存在较大的不确定性和不一致性.对于地震体波走时成像与直流电阻率成像,均面临着成像阴影区问题.对于地震走时成像,地震射线对低速区域覆盖较差形成阴影区,造成低速区域分辨率降低.对于电阻率成像,电场线在高阻区域分布较少,造成高阻区域分辨率较低.为了提高地下介质成像的精度,Gallado和Meju（2003）提出了基于交叉梯度结构约束的联合地球物理成像方法.在要求不同的物性模型拟合各自对应的数据同时,模型之间的结构要求一致,即交叉梯度趋于零.为了更有效地实现基于交叉梯度的结构约束,我们提出了一种新的交替结构约束的联合反演流程,即交替反演不同的数据而且在反演一种数据时要求对应的模型与另一个模型结构一致.新的算法能够更容易地把单独的反演系统耦合在一起,而且也更容易建立结构约束和数据拟合之间的平衡.基于新的联合反演流程,我们测试了基于交叉梯度结构约束的二维跨孔地震走时和直流电阻率联合成像.合成数据测试表明,我们提出的交替结构约束流程能够很好地实现基于交叉梯度结构约束的联合成像.与单独成像结果相比,地震走时和全通道电阻率联合成像更可靠地确定了速度和电阻率异常.