CMG RESEARCH: Combining Adjoint Tomography and Sparse Imaging Methods in Seismology

本项目重点研究地震学反演问题。从地震测量推断地下结构是一个非线性问题，也是一个众所周知的病态问题。pi将通过伴随方法解决该问题的全部非线性，通过迭代最小化变分泛函，其中在每个迭代步骤中，近似解的非线性效应被充分考虑到下一次迭代的计算。为了处理问题的不适定性，他们将使用一种正则化方法，该方法结合了空间分布的有效建模，可以显示不连续以及不连续转换之间的平滑行为。更准确地说，他们将把分布建模为小波和曲线的稀疏叠加，并将相应展开系数的绝对值之和作为惩罚项添加到要最小化的变分泛函中。这一项的加入加强了展开的稀疏性，从而表达了模型在不连续过渡之间的平滑性，并已被证明是正则化的。计算资源已经达到了速度和规模，现在可以将这种方法用于现实问题。地震学试图通过在地表进行测量来深入了解地下地质结构。当振动信号被传送到地下时（无论是通过地震、精心设计的爆炸还是特殊构造的振动器），它们在不同构造的地层中以不同的速度传播，并在不同地层之间的突变中被反射。地震学的目标是从地震仪记录的多重反射及其相互作用产生的复杂信号中重建这些地震波穿过的地下结构。相应的数值问题是如此的复杂，以至于在过去有必要对其进行简化，以保持问题的可行性；这必然导致近似解。pi将利用最近的数学进步，使更有效地模拟异质地下结构成为可能，并利用计算资源速度的持续进步来解决问题，而不必引入以前使用的一些限制性简化。这有望绘制出更精确的地下结构地图。

The focus of this project is on inverse problems in seismography. Inferring undergound structure from seismic measurements is a nonlinear problem that is also notoriously ill-posed. The PIs will tackle this problem in its full nonlinearity via the adjoint method, by iteratively minimizing a variational functional in which, at each iteration step, the nonlinear effects of the approximate solution are fully taken into account for the computations in the next iteration.To deal with the ill-posedness of the problem, they will use a regularization method that incorporates efficient modeling of spatial distributions that can exhibit discontinuities as well as smooth behavior between discontinuous transitions. More precisely, they will model the distribution as a sparse superposition of wavelets and curvelets, and add the sum of the absolute values of the corresponding expansion coefficients as a penalty term to the variational functional to be minimized. The inclusion of such a term enforces the sparseness of the expansion, thus expressing the smoothness of the model between discontinuous transitions, and has been proved to be regularizing. Computational resources have reached the speed and scale at which it is now feasible to use this approach for realistic problems.Seismology seeks to gain insight into underground geological structure by measurements done at the surface. When vibrational signals are sent into the ground (whether by earthquakes, carefully tailored explosions or specially constructed vibrators), they propagate at different speeds through layers of different constitution, and are reflected at the often abrupt transitions between different layers. The goal of seismology is to reconstruct the underground structure traversed by these seismic waves from the complex signals, registered by seismographs, that result from the multiple reflections and their interaction. The corresponding numerical problem is of such great complexity that it has been necessary, in the past, to simplify it so as to keep the problem feasible; this led, of necessity, to approximate solutions. The PIs will make use of recent mathematical advances that make it possible to model more effectively heterogeneous underground structures, and of the continuing progress in speed of computational resources to tackle the problem without having to introduce some of the restrictive simplifications used previously. This is expected to result in more accurate maps of underground structure.