相位空间内的解缠绕相位反演

全波形反演方法是一种数据域高精度反演方法,该方法通过匹配观测数据与模拟数据的地震波形,利用梯度法准确反演地下介质参数的分布情况.由于观测数据普遍缺少低频信息,该方法易受周期跳跃现象影响.特别是当地下存在大尺度强反射界面的构造时,地下介质的反演转化为强非线性问题求解.该情形下,即使观测数据包含充足的低频信息,全波形反演也难以给出准确的反演结果.一般可以通过减弱反演对初始模型参数的依赖性来克服上述问题,具体表现为使用新变量(例如瞬时相位、包络等)代替目标函数中的采样后波场,以增强新目标函数的凸性.但是,对该新目标函数进行反演时,伴随状态方程中存在关于新变量和波场的一个链式微分项,该项保留了反演问题的非线性,导致新的反演方法难以处理包含大尺度构造的强非线性反演问题.此外,基于新变量的反演问题依然在波场空间中计算模型梯度,难以充分利用新变量与模型参数之间的弱非线性关系.因此,本文提出用频率域波动方程的相位形式代替传统的波动方程来消除伴随状态方程中的链式微分项,用解缠绕的相位代替目标函数中采样前波场并在相位空间进行反演.该方法可以最大程度地利用地下介质参数和解缠绕相位之间的弱非线性关系,从而削弱反演的非线性性.由于基于频率域波场计算得到相位有严重的缠绕问题,本文采用基于振幅排序的多聚类算法来对相位进行解缠绕.虽然将介质参数到波场的映射替换为介质参数与解缠绕相位的映射,会导致反演结果的分辨率有所下降,但该方法可以在相位空间恢复介质参数的大尺度低波数分量.Marmousi模型测试证明了该方法的有效性和准确性,针对部分BP模型的测试也证明了该方法处理强非线性问题的能力.

Full unwrapped phase inversion in the phase space

Full waveform inversion (FWI), a high-frequency inversion method in the data domain, often suffers from the cycle-skipping problems for the lack of the low-frequency components in the model. Especially for large-scale strong velocity perturbations, FWI will lose its effectiveness and usually trap in local minimum for its strong non-linear issues, and as a result, fail to converge to reliable inversion results. The common solution to mitigate the reliance on the initial velocity to overcome the cycle-skipping problem is to replace the restricted or sampled wavefield with a new variable, such as the instantaneous phase, envelope and so on. However, the chain differentiation term for the new variable and wavefield of the method above preserve the non-linearity. Therefore, we propose to solve the inversion problem in the phase space by replacing the wavefield in the objective function with the unwrapped phase, which can make the most of the linearity between model parameters and unwrapped phase, and as a result, weakening the non-linearity in the forward operator. We apply the unwrapped algorithm combined with a magnitude-sorted list, multi-clustering algorithm to retrieve the unwrapped phase. The resolution of the inversion results is reduced due to the introduction of the new constraint, which can recover reliable large-scale low wavenumber components in phase space. Numerical tests on Marmousi and BP model with random noise demonstrate the capability of handling the non-linear problem of the proposed method.