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

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.