基于单频走时敏感度核函数的三维波动方程初至层析

3-D wave-equation first-arrival tomography using a monochromatic traveltime sensitivity kernel

地震波走时层析是主要的近地表速度建模技术.对于变速模型,尤其是复杂近地表模型,波动方程能够更加精确地描述地震波的一阶绕射效应,理论上反演精度高于射线(束)类层析.然而,波动方程的数值求解的计算量较射线理论增加了几个数量级,严重制约了波动方程走时层析技术的三维实用化.为解决波动方程走时层析中遇到的计算和存储问题,本文通过引入随机边界条件以及离散Fourier积分,提出了基于单频梯度反演策略的实用化三维波动方程初至层析方法.本文方法的优势在于:在计算和存储方面,相比于基于波场重构算法(需要三次波传播)的梯度计算策略,本文提出的单频梯度计算方法对内存需求极低(相对于波动方程正演,仅需要增加两个单频波场以及一个成像体),且计算量至少减少1/3(仅需要两次波传播且无需处理数值吸收边界).此外,随机边界的引入使得正演算法大幅简化因而更适用于GPU(Graphics Processing Unit,图形处理器)加速.在理论层面,由于影响地震波一阶散射效应的主要区域集中在连接炮检的中心射线邻域的第一菲涅尔带内,随机边界条件的引入降低了随机边界内产生的散射波的空间相关性,使得第一菲涅尔带内的单频梯度相干加强;同时,地震观测系统的多次覆盖特性有助于第一菲涅尔带外的高波数振荡相互干涉.数值实验表明,光滑处理后的单频梯度仍然保留了非常可观的低波数成分,有助于实现背景速度反演.此外,三维复杂模型反演结果验证了本文方法能够较为准确地反演近地表速度结构.相对于传统的波动方程走时层析实现方案,本文方法在显著降低内存需求的同时进一步提高了计算效率,有望将波动方程初至层析技术推向三维实用化.

Seismic traveltime tomography is a key technique for near-surface velocity model building. Specifically, the wave-equation tomography can correctly handle the first-order diffraction phenomena when the scale of velocity heterogeneity is comparable with the incident wavelength. Although wave-equation-based traveltime inversion can theoretically produce a velocity model with higher resolution compared with ray tomography, the computational cost is much higher which prohibits its implementation for industry-scaled 3-D problems. To address this problem, by introducing the random boundary condition and the discrete Fourier integration, we propose a practical 3-D wave-equation first-arrival tomography using the monochromatic gradient inversion scheme. The advantages of the proposed method are: in terms of computational cost and storage space, the proposed method only needs very few memory spaces(i.e., compared with wave-equation forward modeling, only two additional monochromatic wavefields and one image volume are required), and meanwhile, the computational cost is decreased at least by 1/3(i.e., compared with the wavefield reconstruction scheme, only two wavefield propagations are needed, and no special treatment for numerical boundaries). Besides, the introduction of random boundary conditions further simplifies the modeling algorithm which in turn makes it well suited for GPU(Graphics Processing Unit) acceleration. Theoretically, considering that the major area that may influence the first-order diffraction of seismic waves is usually referred to as the first Fresnel zone, which has a very similar shape to the monochromatic sensitivity kernel calculated with the dominant frequency. On one hand, the monochromatic sensitivity kernel within the first Fresnel zone will be coherently increased because the scattered wavefield caused by random boundaries becomes less spatial correlative. On the other hand, the multi-fold seismic acquisition system makes the sensitivity kernels outside the first Fresnel zone further canceled. Numerical example demonstrates that the smoothed monochromatic gradient contains sufficient low-wavenumber components which are adequate for background velocity inversion. Moreover, a 3-D synthetic data test demonstrates that a high-resolution near-surface velocity model can be inverted using wave-equation traveltime tomography with affordable computer resources. Compared with the traditional wave-equation tomography methods, the proposed method not only reduces the memory requirement remarkably but also significantly increases the computational efficiency. Therefore, it is very promising for dealing with industrial-sized 3-D inverse problems.