多源并发下Mur二阶吸收边界和非分裂递归卷积完全匹配层对比研究

多个激励源无延时发射(多源并发)相同中心频率脉冲会形成平面波束信号,增强数据记录质量。本文通过数值模拟对比分析在多源并发情况下,非分裂递归卷积完全匹配层作为吸收边界条件和Mur二阶吸收边界条件对电磁波的吸收效果。其研究结果表明,传统的Mur二阶吸收边界条件对多源并发、多角度掠射情况下电磁波的吸收效果不佳,在大偏移距下会造成波形畸变和形成虚假反射。而在多源并发情况下采用非分裂递归卷积完全匹配层作为吸收边界条件,将坐标伸缩因子引进时域有限差分算法中。通过傅里叶逆变换将频率域坐标伸缩变换PML方程转换到时域,对电场和磁场值在离散状态下进行递归卷积运算求解。从而避免了直接对卷积进行数值求解的复杂计算,在保证计算准确性的同时,节约了内存空间,提高了计算效率。在不分裂波场情况下,改善了网格截断位置对电磁波的吸收效果。     

波动方程显式短算子三维正演模拟

通过对三维波动方程的显式差分法的研究,并对模小于1.0的稳定性约束作线性化处理时引入加权因子,提出了一个基于二次规划方法的一维显式短算子设计方法.与以往的显式算子设计方法的不同之处,在于对传播区内算子的模也引入了最大值限制,这样可更严格地保证深层波场延拓的稳定性.同时,给出了基于一维显式算子及McClellan变换的三维波场深度延拓方法,大大减少了计算量,提高了计算速度,并能更好地处理介质横向非均匀性.     

TTI介质拟声波方程数值模拟

TTI介质qP波数值模拟方法因为考虑了倾角因素,可以比VTI介质qP波数值模拟方法更加准确地描述各向异性介质中地震波场的传播规律。文中用拟声波方程对TTI介质中的地震波场进行了高阶有限差分数值模拟,在改进衰减函数分布方式后,通过坐标变换,利用改进的完全匹配层(perfectly matched layer,PML)边界控制方程对波场边界进行吸收处理,取得了良好的效果;然后分析了拟声波方程数值模拟中的稳定性问题,并对波场中的伪横波进行压制。通过对不同模型的数值模拟,验证了文中使用的TTI介质拟声波波动方程的稳定性以及所采用的PML边界控制方程的可靠性和适用性。     

声波方程隐式高阶有限差分数值模拟

推导了声波方程空间二阶导数的隐式求解公式及差分系数的求解方法,讨论了该方法的数值频散特征。利用该方法分别对均匀介质及Marmousi模型进行了数值模拟,将其结果与传统的显式差分格式的模拟结果进行了对比分析。结果表明：该方法较传统的显式求解方法具有更低的数值频散、更高的计算精度。     

Experiments with Higdon's absorbing boundary conditions for a number of wave equations

Simulation of wave propagation for seismic purposes is usually restricted to a small portion of the earth. Artificial boundary conditions are required where the subsurface model is truncated. Absorbing boundaries should ensure that waves hitting the artificial boundaries are not reflected. The vast amount of literature on the subject suggests that “good” conditions have not been found, and only “reasonable” solutions exist. A cursory overview of existing and a few new ideas is presented that may guide the construction of suitable boundary conditions. Because the intended application of the boundary conditions was a high-order finite-difference code that runs on a parallel computer, we have restricted our attention to local boundary conditions. A fundamental problem in the design of accurate local boundary conditions is pointed out: accuracy is required to keep the amount of reflected energy small, but at the same time allows for growing low-frequency modes. We have settled for Higdon’s boundary conditions. Higdon proposes to include some damping to suppress the growing low-frequency modes. We show that third-order conditions provide acceptable results for the simple scalar wave equation and the acoustic equation. In the elastic case, an additional low-frequency growing mode may occur. This mode can be suppressed by using a dissipative boundary scheme and by increasing the amount of damping. The increase in damping results in an increase in the amount of reflected energy, which is larger than in the scalar case. Numerical experiments exhibit a reasonable performance, although some improvement would be useful, particularly in the anisotropic elastic case. This revised version was published online in July 2006 with corrections to the Cover Date.     

Aerodynamic entrainment threshold: effects of boundary layer flow conditions

Experimental data are presented demonstrating the influence of boundary layer flow conditions on aerodynamic entrainment of grains in the absence of intersaltation collisions. New methods are proposed for (1) the unambiguous determination of aerodynamic threshold for any grain population and (2) approximation of the probability density function (PDF) distributions of threshold shear velocity for aerodynamic entrainment. In wind tunnel experiments, the orderly spatial development of flow conditions within a developing boundary layer over the roughened surface of a flat plate constrains the aerodynamic threshold condition in terms of both mean and fluctuating values. Initial grain dislodgements and subsequent erosion from narrow strips of loose, finely fractionated ballotini were recorded photographically as wind speed was increased. Boundary layer parameters, including average threshold shear velocity (U_*t), were calculated using the momentum integral method. Direct observations show that sporadic oscillation of grains preceded dislodgement. At slightly higher velocities most grains rolled over their neighbours before entering saltation. Initial entrainment in spatially semi-organized flurries of 50 or more grains was followed by quiescent periods at airflow velocities close to threshold. These observations provide strong circumstantial evidence linking both the nature and spatial pattern of initial grain motions to sweep events during the fluid bursting process. For each grain fraction, values of U_*t were found to span an unexpectedly wide range and to decrease downwind from the leading edge of the plate as turbulence intensity increased. A probabilistic entrainment model is applied to the aerodynamic threshold condition so as to incorporate the effects of changing turbulent flow regimes over the plate. Analysis of strip erosion curves gives both an objective definition of the threshold condition and usable approximations of the PDF for U_*t required by the model and for future stochastic treatment of the threshold condition.     

求解声波方程的径向基函数无网格法

地震勘探广泛应用于油气、煤田勘探。地震波场数值模拟是整个地震勘探数据处理技术的基石。将径向基函数（RBF）引入地震声波波场数值模拟中,在空间上用径向基函数无网格法来构造二阶导数,而在时间上采用简单的二阶差分公式,并重点讨论了形状参数c对该方法精度的影响,总结c经验取值范围为2~4倍平均数据点间距。设计不同模型,利用径向基函数无网格法进行声波波场模拟,并与空间四阶时间二阶的有限差分计算结果进行对比,结果表明：同样精度下,径向基函数每个波长所取的数据点数远小于空间四阶矩形网格有限差分每个波长所取的网格点数,即径向基函数的空间采样率更低,这表明径向基函数具有更小的数值频散。     

Mixed unsplit-field perfectly matched layers for transient simulations of scalar waves in heterogeneous domains

We discuss a new formulation for transient scalar wave simulations in heterogeneous semi-infinite domains. To deal with the semi-infinite extent of the physical domains, we introduce truncation boundaries and adopt perfectly matched layers (PMLs) as the boundary wave absorbers. Within this framework, we develop a new mixed displacement-stress (or stress memory) finite element formulation based on unsplit-field PMLs. We use, as typically done, complex-coordinate stretching transformations in the frequency domain, and recover the governing partial differential equations in the time-domain through the inverse Fourier transform. Upon spatial discretization, the resulting equations lead to a mixed semi-discrete form, where both displacements and stresses (or stress histories/memories) are treated as independent unknowns. We propose approximant pairs, which, numerically, are shown to be stable. The resulting mixed finite element scheme is relatively simple and straightforward to implement, when compared against split-field PML techniques. It also bypasses the need for complicated time integration schemes that arise when recent displacement-based formulations are used. We report numerical results for 1D and 2D scalar wave propagation in semi-infinite domains truncated by PMLs. We also conduct parametric studies and report on the effect the various PML parameter choices have on the simulation error.     

Influence of rigid boundary on the propagation of torsional surface wave in an inhomogeneous layer

Journal of Earth System Science - The present work illustrates a theoretical study on the effect of rigid boundary for the propagation of torsional surface wave in an inhomogeneous crustal layer...     

Consistent transmitting boundary conditions with a reduced number of eigenmodes for wave propagation in elastic media

The consistent transmitting boundary condition is a powerful numerical technique for frequency domain analysis of wave problems in unbounded elastic media. A second-order complex-valued eigen-problem is associated with this method and the computed eigenmodes are used to formulate the nodal forces acting on the lateral boundaries of a Cartesian computational domain. Usually, all eigenmodes are computed and used in the consistent transmitting boundary conditions to simulate the infinite extent of the elastic medium. In this paper, it is demonstrated that by selecting a reduced number of relevant eigenmodes it is possible to produce similar results and consequently reduce the overall computational cost. This is highly desirable for soil–structure interaction problems involving wide ranges of frequencies and large-size domains with many sub-layers. The efficiency of the proposed approach is investigated by considering, first, the case of foundation vibration over rigid bedrock and, second, the effectiveness of a wave barrier to reduce the transmitted vibrations caused by a surface harmonic loading.     

非均匀层状介质一维波动方程精确解的有限差分算法

平面波的传播问题通常可以归结为一维波动方程的定解问题。在非均匀介质中,即使简单的一维波动方程也需要借助于数值方法获得近似解。3层5点古典差分格式是计算偏微分方程一种常用算法,作为一种显式迭代格式,需要满足稳定性条件 ,其中 为波速, 为空间采样间隔, 为时间采样间隔。当 时, ,古典差分格式达到临界稳定状态。在这种情况下,平面波在 时间内的传播距离恰好等于空间采样间隔,差分格式真实地反映了平面波的传播原理,因而可以得到一维波动方程的精确解。但是,由于在非均匀介质中存在不连续的波阻抗界面,此方法不适于计算非均匀介质的波场。为了将临界稳定情况下的古典差分格式推广应用至非均匀层状介质,提出了一种能够处理波阻抗界面的有限差分格式,并应用傅里叶分析法得到其稳定性条件。模型算例验证了此算法的正确性。     

An MSSS-preconditioned matrix equation approach for the time-harmonic elastic wave equation at multiple frequencies

In this work, we present a new numerical framework for the efficient solution of the time-harmonic elastic wave equation at multiple frequencies. We show that multiple frequencies (and multiple right-hand sides) can be incorporated when the discretized problem is written as a matrix equation. This matrix equation can be solved efficiently using the preconditioned IDR(s) method. We present an efficient and robust way to apply a single preconditioner using MSSS matrix computations. For 3D problems, we present a memory-efficient implementation that exploits the solution of a sequence of 2D problems. Realistic examples in two and three spatial dimensions demonstrate the performance of the new algorithm.     

频率域单程方程三维叠前正演中的波场替换

利用定位原理进行三维叠前正演,仅靠单程声波方程的延拓即可形成复杂地质条件下的正演记录,运算效率高无多次反射出现。但速度模型中对于过大或过小的速度常使波场在延拓时有频率和波数成分的损失,这种损失引起波场畸变并严重影响正演记录的质量。本文采用波场替换技术,在每次延拓时用零相位子波形成一个同时保留了原波场的动力学特征与空间位置信息的新波场,然后用新波场取代原畸变波场并参与正演记录合成。在频率域单程方程三维叠前正演中特别适合做波场替换,因为震源脉冲与接收响应(均可视为点脉冲)的延拓波场在形态上相似,而且在延拓波场中每道记录之振幅包络只有一个极值点。理论试算结果表明,采用波场替换技术后最终正演记录质量比替换前有了很大改善。     

A note on indirect boundary element methods for the diffusion equation

This paper corrects an impression, created by a recent contribution to the International Journal for Numerical and Analytical Methods in Geomechanics, that the mathematical equivalence of the direct and indirect boundary element methods for the diffusion equation implies that their respective computational needs with regard to the body integral terms are equal. It is shown that in a stepwise implementation of the numerical procedure for solving a finite body problem, indirect methods requirean integration over an infinite region and not merely over the body as is the case for the direct version.     

Method and analysis for the upscaling of structural data

3D geological models are created to integrate a set of input measurements into a single geological model. There are many problems with this approach, as there is uncertainty in all stages of the modelling process, from initial data collection to the approach used in the modelling scheme itself to calculate the geological model. This study looks at the uncertainty inherent in geological models due to data density and introduces a novel method to upscale geological data that optimises the information in the initial dataset. This method also provides the ability for the dominant trend of a geological dataset to be determined at different scales. By using self-organizing maps (SOM's) to examine the different metrics used to quantify a geological model, we allow for a larger range of metrics to be used compared to traditional statistical methods, due to the SOM's ability to deal with incomplete datasets. The classification of the models into clusters based on the geological metrics using k-means clustering provides a useful insight into the models that are most similar and models that are statistical outliers. Our approach is guided and can be calculated on any input dataset of this type to determine the effect that data density will have on a resultant model. These models are all statistical derivations that represent simplifications and different scales of the initial dataset and can be used to interrogate the scale of observations.     

Thermal and wind structure of the monsoon trough boundary layer

Radiosonde data from Jodhpur, taken at 0530, 1730 and around 1100 hr IST during MONTBLEX 1990, reveal that the distribution of virtual potential temperature0 _v below about 500 hPa has a structure characterized by up to three layers each of approximately constant gradient. We are thus led to introduce a characterization of the observed thermal structure through a sequence of the symbolsN, S andU, standing respectively for neutral, stable or unstable conditions in the different layers, beginning with the one closest to the ground. It is found that, of the 29 combinations possible, only the seven classes,S, SS′, SNS′, NS, NSS′, USS′ andUNS are observed, whereS′ stands for a stable layer with a different gradient of0 _r. than in the layerS. It is also found that, in 90% of the launches at 0530 hr, 48% of the launches at 1730 hr and 69% of the launches around 1100 hr, the first radiosonde layer near the ground is stable; the classical mixed layer was found in only 11 % of the data set analysed, and, if present on other occasions, must have been less than 250 m in height, the first level at which radiosonde data are available. Supplementing the above data, sodar echograms, available during 82% of the time between June and August 1990, suggest a stable layer up to a few tens of metres 48% of the time. A comparative study of the radiosonde data at Ranchi shows that the frequent prevalence of stability near the surface at Jodhpur cannot be attributed entirely to the large scale subsidence known to be characteristic of the Rajasthan area. Further, data at Jodhpur reveal a weak low level jet at heights generally ranging from 400 to 900 m with wind speeds of 6 to 15 m/s. Based on these results, it is conjectured that the lowest layers in the atmosphere during the monsoons, especially with heavy clouding or rain, may frequently be closer to the classical nocturnal boundary layer than to the standard convective mixed layer, although often with shallow plumes that penetrate such a stable layer during daytime.     

Melilitites: partial melts of the thermal boundary layer?

Silica-poor, calcium-rich melilitites form a chemical and isotopic end-member of the spectrum of mafic magmas of the Tertiary-Quaternary volcanic province of western and central Europe. We propose that these unusual magmas are derived by partial melting of the thermal boundary layer (TBL) at the base of the European lithosphere. The processes involved in the evolution of the TBL have been constrained using major and trace element and Nd-Sr-Pb isotope data for Tertiary melilitites from the Urach, Hegau, and Rhine graben regions of Germany. The initiation of boundary layer evolution is limited in time by a major phase of Permo-Carboniferous rifting and associated magmatism postdating the Hercynian orogeny which would have destroyed the existing TBL by delamination. Model calculations indicate that the isotopic composition of the melilitite source cannot develop within the TBL over geologically reasonable periods of time (250–300 Million years) if the TBL evolves solely by incorporation of small degree (<0.1%) partial melts from an underlying, convecting, depleted (MORB-source) mantle reservoir. On the basis of this observation, the regional geodynamic setting and the melilitite data, we propose that an isotopically distinct mantle plume, impinging on the base of the European lithosphere during the Early Cenozoic, is involved in the petrogenesis of the melilitite magmas.     

单程波法地震波衰减特征数值模拟分析

在非均匀粘弹性介质中,地震波衰减研究能够有效揭示地层吸收特性对地震波的影响。这里从单程波方程出发,结合粘弹性介质中的复波数理论,考虑介质的横向变化,采用分步傅立叶波场延拓技术,推导出适用于横向变速介质的带Q地震记录计算公式,最后将此方法应用于四个不同含Q地层模型。试验表明：单程波法计算速度快,波场信息简单清晰,易于深入分析和反演算法的建立。并且,Q值对地震波波形、能量、频率等都产生了较为明显的影响,对实际资料能量补偿、地震资料分辨率的提高,以及利用衰减特征进行储层识别有一定指导价值。     

A simple boundary equation technique for elastic stress analysis of underground cavities

Summary A boundary integral equation approach is illustrated which does not require the solution of singular integrals in order to compute the coefficients of the final system of linear equations. This characteristic allows the carrying out of all the necessary integrations by means of numerical techniques, leading to a rather simple and compact solution algorithm. In addition the approach permits the evaluation of a measure of the error affecting the numerical solution. The stability and the accuracy of the procedure, which is based on the minimization of the difference between known boundary tractions and corresponding numerically derived stresses, are discussed with reference to some rock engineering problems and it is shown that the approach is particularly suitable for stress analyses related to underground openings. To get some insight into the overall performance of the procedure, the results obtained in the analysis of a tunneling problem of practical interest are compared with those deriving from a standard Finite Element calculation.