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

这个项目的重点是地震学中的反问题。 从地震测量推断地下结构是一个非线性问题，也是众所周知的不适定。PI将通过伴随方法在完全非线性的情况下处理这个问题，通过迭代地最小化变分泛函，在每个迭代步骤中，近似解的非线性影响在下一次迭代的计算中被充分考虑。为了处理问题的不适定性，他们将使用一种正则化方法，该方法结合了对空间分布的有效建模，这些空间分布可以表现出不连续性以及不连续过渡之间的平滑行为。更准确地说，他们将分布建模为小波和曲波的稀疏叠加，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。包含这样一个项加强了扩展的稀疏性，从而表达了模型在不连续过渡之间的平滑性，并且已被证明是正则化的。 计算资源的速度和规模已经达到了现在使用这种方法解决现实问题的可行性。地震学试图通过在地面进行测量来深入了解地下地质结构。当振动信号被发送到地面时（无论是通过地震，精心设计的爆炸还是特殊构造的振动器），它们以不同的速度通过不同构造的层传播，并在不同层之间的突然过渡处反射。地震学的目标是从地震仪记录的复杂信号中重建这些地震波穿过的地下结构，这些信号是由多次反射及其相互作用产生的。相应的数值问题是如此之大的复杂性，它一直是必要的，在过去，以简化它，以保持问题的可行性;这导致，必要的，近似的解决方案。PI将利用最新的数学进步，使其能够更有效地模拟异质地下结构，并在计算资源的速度方面不断取得进展，以解决这个问题，而不必引入以前使用的一些限制性简化。预计这将导致更准确的地下结构地图。

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.