高斯波束-Born散射波模拟实现技术与基于复程函方程的地震波复走时计算

发布时间：2018-01-16 20:16

本文关键词：高斯波束-Born散射波模拟实现技术与基于复程函方程的地震波复走时计算　出处：《吉林大学》2017年硕士论文　论文类型：学位论文

【摘要】：高斯波束-Born理论是基于高斯波束叠加表示格林函数的高斯波束层析成像、全波形反演以及广义绕射叠加理论的基础。其数学表达式是一种体积分形式,由此而产生的实现方案为对整个散射区域的所有散射场的逐次叠加。为了解决Born积分中体积分计算量大的问题,本文将高斯波束-Born公式中的体积分变为面积分,并用一种改进的等时面叠加方案完成计算,这种实现方式的优点是能够提高计算效率。此外,高斯波束-Born方法的基础是高斯波束计算,对于高斯波束的计算主要包含振幅和相位两部分,其中,振幅计算是依赖于射线追踪算法,而相位部分是角频率和复走时的乘积。目前,对于复走时的计算主要是通过给定复值初值的动力学射线追踪法。但是,动力学射线追踪法依赖于傍轴近似,复走时数学表达式是以二阶Taylor展开为基础,并且该表达式只运用到中心射线上的速度,而没有顾及射线两侧的速度变化。因此,当波束传播到速度强烈不均匀处,特别是中心射线两侧速度不对称时,复走时计算精度收到很大影响,从而限制了高斯波束的适用范围。一种运用扰动理论建立的高斯牛顿-快速推进法,通过直接对复程函方程进行求解来计算复走时能解决上述困难。但是,在应用上,该方法具有两个弱点:1)目前只能运用于简单的垂直射线计算,而无法计算一般情况下的弯曲射线复走时和高斯波束;2)在对虚慢度的计算中,由于运用了最优化方法,因此计算量较大,计算效率受到影响。为了克服上述弱点,本文将非等距网格差分法引入处理弯曲中心射的弯曲边界问题,提出了能适用于弯曲射线的基于复程函方程的复走时计算方法。同时,将L-BFGS优化方法引入到计算等效虚慢度中,该方法在计算过程中避免了多次计算和储存Hessian矩阵,数值实例表明,提高了复走时计算效率。根据定义,用以描述这种具有衰减特性的波的复走时必须依附于所考虑的射线,即本文计算的复走时是一种局部复走时。具体地,局部复走时实部和虚部分别为常规射线走时和傍轴走时。因此,在对单个射线的复走时计算时,如果不对傍轴区域加以限制,将计算整个模型的复走时。为了计算具有一定范围的局部复走时,根据具体应用需要,提出了局部算法计算局部复走时,数值结果证明了其适用性。与上述求解各向同性介质中复程函方程不同,各向异性介质复程函方程涉及参数众多。因此,对于各向异性介质(例如TTI介质)复走时计算时,上述方法受到很大限制。为此,本文利用扰动理论思想,提出了将TTI复程函方法线性化从而计算复走时。数值结果表明,扰动方法对求解TTI复程函方程是有效的。最后,通过对比发现,根据上述复走时计算方法计算的复走时进一步计算的高斯波束波场相比于动力学射线追踪方法计算结果在精度上很大改善。
[Abstract]:Gao Si's beam-Born theory is the basis of Gao Si beam tomography, full waveform inversion and generalized diffraction superposition theory based on the Gao Si beam superposition representation. The mathematical expression of Gao Si beam superposition theory is a form of volume division. The resulting scheme is the successive superposition of all scattering fields in the whole scattering region, in order to solve the problem of large volume integral calculation in Born integral. In this paper, the volume fraction of Gao Si beam Born formula is changed into area fraction, and a modified isochronous superposition scheme is used to complete the calculation. The advantage of this realization is that the calculation efficiency can be improved. Gao Si beam-Born method is based on Gao Si beam calculation. The calculation of Gao Si beam mainly includes two parts: amplitude and phase, in which amplitude calculation is dependent on ray tracing algorithm. The phase part is the product of the angular frequency and the complex travel time. At present, the calculation of the complex travel time is mainly based on the dynamic ray tracing method with given complex initial values. However, the dynamic ray tracing method depends on the paraxial approximation. The mathematical expression of complex walk time is based on the second-order Taylor expansion, and the expression is applied only to the velocity of the central ray, without taking into account the variation of the velocity on both sides of the ray. When the beam propagates to the strong inhomogeneity of velocity, especially when the velocity of the two sides of the central ray is asymmetrical, the calculation accuracy of the complex beam is greatly affected. Thus limiting the scope of application of Gao Si beam. By solving the complex function equation directly to calculate the difficulties mentioned above, a rapid advance method based on the perturbation theory is proposed. However, the above problems can be solved by solving the complex function equation directly. However, this method can be used to solve the problems mentioned above. In application, the method has two weaknesses: 1) at present, it can only be used in simple vertical ray calculation, but can not calculate the bending ray rerun time and Gao Si beam in general. 2) in the calculation of virtual slowness, due to the use of optimization method, the computational complexity is large and the computational efficiency is affected. In this paper, the nonequidistant grid difference method is introduced to deal with the bending boundary problem of bending center ejection, and a complex travel time calculation method based on complex function equation is proposed, which can be applied to bending rays. The L-BFGs optimization method is introduced into the calculation of equivalent virtual slowness. The method avoids calculating and storing the Hessian matrix many times in the process of calculation. According to the definition, the rerun time of this kind of wave with attenuation characteristic must be attached to the ray under consideration, that is, the calculated rerun time in this paper is a kind of local rerun time. The real part and the imaginary part are regular ray travel time and paraxial travel time, respectively. Therefore, if the paraxial region is not restricted in the calculation of the rerun time of a single ray. In order to calculate the local rerun time of the whole model, a local algorithm is proposed to calculate the local rerun time according to the specific application needs. The numerical results prove its applicability. Different from solving the complex equation in isotropic medium, the complex equation of anisotropic medium involves many parameters. For anisotropic media (such as TTI medium), the above method is limited greatly. Therefore, the perturbation theory is used in this paper. The TTI complex function method is linearized to calculate the complex travel time. The numerical results show that the perturbation method is effective for solving the TTI complex function equation. The accuracy of the further calculation of the Gao Si beam wave field based on the above complex travel time calculation method is much better than that of the dynamic ray tracing method.
【学位授予单位】：吉林大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：P631.4

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