标题： 理解地震学中的伴随方法：时域中的理论与实现 显示英文标题

标题： Understanding the Adjoint Method in Seismology: Theory and Implementation in the Time Domain

摘要： 伴随方法是当今用于地震（全波形）反演的一种常用方法。 该方法通过使用更真实的物理模型，被认为能够提供更真实和详细的地球内部图像。 它依赖于一个伴随波场的定义（因此得名），这个波场是时间反转的合成数据，满足原始运动方程。 然而，伴随波场性质的物理解释通常通过蛮力方法、临时假设和/或依赖格林函数的存在、表示定理和/或Born近似来完成。 我们仅使用变分原理，不依赖上述假设和/或额外的数学工具，证明时间反转的伴随波场应被定义为导致正确伴随方程的前提。 这使我们能够澄清在处理运动方程中的粘弹性衰减和/或奇数阶导数项时，在以前的开创性工作中发现的数学不一致之处。 然后我们讨论该方法在时域中的数值实现方法，并提出一种变分公式，用于构建不同的误差函数。 我们在这里定义了一个新的误差旅行时间函数，使我们能够就交叉相关旅行时间测量所显示的沿射线路径的零灵敏度这一长期争论达成共识。 事实上，我们证明沿射线路径的零灵敏度是进行交叉相关旅行时间测量所需的数据与合成数据之间相似性假设的结果。 当没有预先设定数据与合成数据之间的假设时，旅行时间Frechet核在射线路径上表现出极值，这与直觉预期一致。 显示英文摘要

摘要： The adjoint method is a popular method used for seismic (full-waveform) inversion today. The method is considered to give more realistic and detailed images of the interior of the Earth by the use of more realistic physics. It relies on the definition of an adjoint wavefield (hence its name) that is the time reversed synthetics that satisfy the original equations of motion. The physical justification of the nature of the adjoint wavefield is, however, commonly done by brute force with ad hoc assumptions and/or relying on the existence of Green's functions, the representation theorem and/or the Born approximation. Using variational principles only, and without these mentioned assumptions and/or additional mathematical tools, we show that the time reversed adjoint wavefield should be defined as a premise that leads to the correct adjoint equations. This allows us to clarify mathematical inconsistencies found in previous seminal works when dealing with visco-elastic attenuation and/or odd-order derivative terms in the equation of motion. We then discuss some methodologies for the numerical implementation of the method in the time domain and to present a variational formulation for the construction of different misfit functions. We here define a new misfit travel-time function that allows us to find consensus for the long-standing debate on the zero sensitivity along the ray path that cross-correlation travel-time measurements show. In fact, we prove that the zero sensitivity along the ray-path appears as a consequence of the assumption on the similarity between data and synthetics required to perform cross-correlation travel-time measurements. When no assumption between data and synthetics is preconceived, travel-time Frechet kernels show an extremum along the ray path as one intuitively would expect.