The adjoint method in seismology—: II. Applications: traveltimes and sensitivity functionals

Sensitivity functionals which allow us to express the total derivative of a physical observable with respect to the model parameters, are defined on the basis of the adjoint method. The definition relies on the existence of Green’s functions for both the original and the adjoint problem. Using the acoustic wave equation in a homogeneous and unbounded medium, it is shown that the first derivative of the wavefield u with respect to the model parameter p = c − 2 (c is the wave speed) does not contain traveltime information. This property also influences objective functions defined on u and in particular the least squares objective function. Therefore, a waveform inversion should either be complemented by a traveltime tomography or work with initially very long wavelengths that decrease in the course of the iteration. The definition of the sensitivity functionals naturally introduces waveform sensitivity kernels. Analytic examples are shown for the case of an isotropic, elastic and unbounded medium. In the case of a double couple source there are three classes of sensitivity kernels, one for each of the parameters λ , μ (Lamé parameters) and ρ (density). They decompose into P → P, P → S, S → P and S → S kernels and can be described by third-order tensors incorporating the radiation patterns of the original wavefield and the adjoint wavefield. An analysis of the sensitivity kernels for density suggests that a waveform inversion procedure should exclude either the direct waves or the source and receiver regions. S-wave sensitivity kernels, corresponding to conversions from or to S-waves, are larger than the kernels corresponding to P-waves only. This implies that S-wave residuals caused by parameter differences between the true Earth model and the numerical model will dominate a waveform inversion that does not account for that effect.