Multi-parameter full-waveform inversion for VTI media based on Born sensitivity kernels
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摘要: 本文基于VTI介质拟声波方程,利用散射积分原理,在Born近似下导出了速度与各向异性参数的敏感核函数,同时结合作者前期研究提出的矩阵分解算法实现了一种新的VTI介质多参数全波形反演方法.矩阵分解算法通过对核函数-向量乘进行具有明确物理含义的向量-标量乘分解累加运算实现目标函数一阶方向或二阶方向的直接求取,从而避免了庞大核函数矩阵与Hessian矩阵的存储,该方法同时可以大大降低常规全波形反演在计算二阶方向时的庞大计算量.为了克服不同参数对波场影响程度的不同,本文利用作者前期在VTI介质射线走时层析成像研究中提出的分步反演策略实现了多参数联合全波形反演.理论模型实验表明,本文提出的基于Born敏感核函数的各向异性矩阵分解全波形反演方法可以获得较好的多参数反演结果.Abstract: Anisotropic full waveform inversion (FWI) is being widely studied. However, it remains a problem for FWI to reveal all the anisotropic parameters simultaneously because of their coupling. Parameterization and inversion strategy are important solutions to such a problem. In this study, we propose a new VTI FWI method based on a pseudo-acoustic wave equation. This method uses a matrix decomposition algorithm to explicitly construct the gradient of the objective function instead of the traditional adjoint-state method. Meanwhile, a triple-round strategy is tested effective to implement the joint inversion of all the three parameters.In FWI, the least-squares objective function is usually used, and the increment Δm=(Δv,Δε,Δδ) around a starting model m0=(v0,ε0,δ0) can be expressed as the product of a direction vector p and a step length t under local optimization theory. The direction vector p equals the product of the transposed kernels K and the wavefield residual. Based on frequency-domain pseudo-acoustic wave equations in VTI media, we derive the corresponding sensitivity kernels for the three parameters v,ε,δ. However, the kernels are too huge to store in memory. Therefore, we use a matrix decomposition algorithm, instead of the adjoint-state method, to calculate p and t by accumulation of single vector-scalar products. By this way, the huge kernel matrix K is not necessarily stored in memory beforehand. In order to obtain VTI parameters simultaneously, we analyze the sensitivities and find that v has the most dominant influence on the wavefield, δ is the weakest, and ε is in the middle. Therefore, we use a triple-round strategy to reveal the three parameters. In the first round, we invert simultaneously for (v1,ε1,δ1) according to an initial model (v0,ε0,δ0). In the second round, we use (v1,ε0,δ0) as the initial model and invert simultaneously for (v2,ε2,δ2). In the third round, we use v2,ε2,δ0 as the initial model and invert simultaneously for (v3,ε3,δ3). By this way, both the strong and weak parameters can be successfully constructed.In order to test the inversion capability of this method, we design a 2D VTI homogenous model with three anomalies inside. In this experiment we only invert for v and ε simultaneously, with δ being fixed as the true model because δ has a very weak influence on data. The dimension of this model is 4 km×4 km, with the discretized interval 10 m×10 m. In total 52 shots are uniformly distributed along the shot side, and 60 receivers are uniformly laid along the other three sides away from the shot side. Inversion is accomplished in the frequency domain. The starting frequency is 2 Hz, and the end frequency is 11.5 Hz, with frequency interval 0.5 Hz. The starting model of v is homogeneous without anomalies. The starting model of ε is a highly smoothed version of the true ε model. Finally, both v and ε are well revealed, while the resolution of obtained ε is not as high as v because of its natural weaker influence on the wavefield. We propose a new FWI scheme for VTI media based on Born kernels derived from a pseudo-acoustic wave equation. A matrix decomposition algorithm is employed to calculate the directions and step length through accumulation, while without need to store the huge Fréchet kernel or the approximate Hessian beforehand. To reveal all the anisotropic parameters, we use a triple-round strategy to overcome the different influences of different parameters on the wavefield. Numerical experiment proves the effectiveness of this method and shows its potential to construct all the parameters in complex anisotropic media.
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