Projection methods and applications for seismic nonlinear inverse problems with multiple constraints
Projection methods and applications for seismic nonlinear inverse problems with multiple constraints
复制标题
多约束地震非线性反问题的投影方法及应用
作者：
B. Peters;B. Smithyman;F. Herrmann
非线性反问题常常受到局部极小值的阻碍，因为数据中缺少低频和远偏移，缺乏良好的初始模型，噪声和建模误差。一个众所周知的方法来对付这些不足之处是包括先验信息的未知模型，这正则化的逆问题。虽然传统的正则化方法在不适定（地球物理）反问题中取得了巨大的进展，但当先验信息由多个部分组成时，挑战仍然存在。为了处理这种情况，我们开发了一个优化框架，允许我们以约束的形式添加多条先验信息。所提出的框架是更适合于全波形反演（FWI），因为它提供了保证，多个约束是唯一地施加在每次迭代，无论它们被调用的顺序。为了唯一地投影到多个集合的交集上，我们使用不依赖于权衡参数的Dykstra算法。从这个意义上说，我们的方法与Tikhonov/惩罚正则化和梯度滤波等方法有很大不同。这些都没有提供保证，这使得它们不太适合FWI，不切实际的中间结果有效地破坏了反演。通过处理集合的交集，我们避免了权衡参数，并将客观计算与投影分开，这些投影通常比3D中的目标/梯度计算快得多。这些功能允许轻松集成到现有的代码库中。使用约束还允许进行简化，在这种情况下，我们通过逐渐放松约束来建立模型的复杂性。这种策略有助于避免收敛到代表不切实际的模型的局部最小值。使用多个约束条件，我们得到更好的FWI结果相比，二次惩罚方法，而所有的约束条件的定义是在物理单位，直接从先验知识。
Nonlinear inverse problems are often hampered by local minima because of missing low frequencies and far offsets in the data, lack of access to good starting models, noise, and modeling errors. A well-known approach to counter these deficiencies is to include prior information on the unknown model, which regularizes the inverse problem. Although conventional regularization methods have resulted in enormous progress in ill-posed (geophysical) inverse problems, challenges remain when the prior information consists of multiple pieces. To handle this situation, we have developed an optimization framework that allows us to add multiple pieces of prior information in the form of constraints. The proposed framework is more suitable for full-waveform inversion (FWI) because it offers assurances that multiple constraints are imposed uniquely at each iteration, irrespective of the order in which they are invoked. To project onto the intersection of multiple sets uniquely, we use Dykstra’s algorithm that does not rely on trade-off parameters. In that sense, our approach differs substantially from approaches, such as Tikhonov/penalty regularization and gradient filtering. None of these offer assurances, which makes them less suitable to FWI, where unrealistic intermediate results effectively derail the inversion. By working with intersections of sets, we avoid trade-off parameters and keep objective calculations separate from projections that are often much faster to compute than objectives/gradients in 3D. These features allow for easy integration into existing code bases. Working with constraints also allows for heuristics, where we built up the complexity of the model by a gradual relaxation of the constraints. This strategy helps to avoid convergence to local minima that represent unrealistic models. Using multiple constraints, we obtain better FWI results compared with a quadratic penalty method, whereas all definitions of the constraints are in terms of physical units and follow from the prior knowledge directly.