On the steadfastness of the least-squares reverse-time migration wavefield extrapolation via 1st-order Riemannian axis finite-difference solver

Hussein Muhammed , AbdelHafiz Gadelmula , Zhenchun Li
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Abstract

Exploring Earth's deep regions via Least-Squares Reverse-Time Migration (LSRTM) methods is of significant interest due to its exceptional structural-amplitude clarity. This cutting-edge seismic imaging technique is time-consuming and memory-intensive, thus wavefield extrapolation was proposed to be in the Pseudodepth domain (1st-order Riemannian coordinate system’s axis) to address these issues and to prevent oversampling/aliasing when modeling deeper subsurface zones. Stabilizing the generated Riemannian wavefield involves implementing an appropriate mapping velocity and obtaining the vertical axis operator which partially converts the finite difference solver from time to frequency domains. Each Cartesian point (x,y,z) has a corresponding vertical-time point (ξ1,ξ2,ξ3), allowing interpolation of the reconstructed source wavefield through a Cartesian-to-Riemannian mapping function. Our stability and convergence analysis indicates that the spatial derivatives of the 1st-order Riemannian axis can be approximated by Fourier pseudo-spectral methods and fast-Fourier transforms using a special Gaussian-like impulse function. This function generates the source term vector-matrix within the finite-difference operator. The mapping velocity, derived as a differential form of the initial input velocity model, controls the CFL conditions of the associated Riemannian-finite difference operator. Numerical, synthetic, and seismic field data examples show that this approach is more stable and efficient in extrapolating a smooth 1st-order Riemannian axis-based finite-difference wavefield while adhering to Claerbout’s principle for locating subsurface reflectors. Additionally, choosing the appropriate sampling rate for the new vertical axis is inversely related to the maximum frequency of the impulse wavelet and directly related to the minimum velocity value in the model.
一阶黎曼轴有限差分解算器最小二乘逆时偏移波场外推的稳定性
通过最小二乘逆时偏移(LSRTM)方法探测地球深部区域由于其特殊的结构振幅清晰度而引起了极大的兴趣。这种尖端的地震成像技术耗时且内存密集,因此提出在伪深度域(一阶黎曼坐标系的轴)进行波场外推,以解决这些问题,并防止在更深的地下区域建模时过采样/混叠。稳定生成的黎曼波场包括实现适当的映射速度和获得垂直轴算子,该算子将有限差分求解器部分地从时域转换为频域。每个笛卡尔点(x,y,z)都有对应的垂直时间点(ξ1,ξ2,ξ3),可以通过笛卡尔-黎曼映射函数对重构的源波场进行插值。我们的稳定性和收敛性分析表明,一阶黎曼轴的空间导数可以用傅里叶伪谱方法和使用特殊的类高斯脉冲函数的快速傅里叶变换来近似。这个函数在有限差分算子内生成源项向量矩阵。映射速度,作为初始输入速度模型的微分形式,控制相关黎曼-有限差分算子的CFL条件。数值、合成和地震现场数据实例表明,该方法在外推光滑的一阶黎曼轴有限差分波场时更加稳定和有效,同时坚持了Claerbout定位地下反射器的原理。此外,为新的纵轴选择合适的采样率与脉冲小波的最大频率成反比,与模型中的最小速度值直接相关。
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