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 viscoelastic 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 longstanding 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 Fréchet kernels show an extremum along the ray path as one intuitively would expect.

伴随方法是当今用于地震（全波形）反演的流行方法。该方法被认为通过使用更真实的物理学来提供更真实和详细的地球内部图像。它依赖于伴随波场（因此得名）的定义，伴随波场是满足原始​​运动方程的时间反转合成。然而，伴随波场的性质的物理证明通常是通过具有临时假设的强力和/或依赖于格林函数、表示定理和/或玻恩近似的存在来完成的。仅使用变分原理，并且没有这些提到的假设和/或额外的数学工具，我们表明时间反演伴随波场应该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先前的开创性工作中发现的数学不一致之处。然后，我们讨论了在时域中数值实现该方法的一些方法，并提出了构造不同失配函数的变分公式。我们在这里定义了一个新的失配走时函数，使我们能够就互相关走时测量显示的沿射线路径的零灵敏度的长期争论达成共识。事实上，我们证明，沿着射线路径出现的零灵敏度是由于执行互相关走时测量所需的数据和合成数据之间的相似性的假设的结果。 当数据和合成之间没有预先假设时，走时 Fréchet 核会按照人们直观的预期沿着光线路径显示一个极值。