Waveform Inversion in the Image Domain

Waveform inversion is a costly but accurate technique for model building. This method places huge demands on the data by requiring extremely low frequencies to reduce cycle skipping. One way of addressing this problem is to use an objective function based on correlations instead of differences between the observed and simulated wavefields. This procedure reduces cycle skipping, although it disregards information carried by the wavefield amplitudes. Correlation-based waveform inversion can be implemented either in the data or image domains. The key elements of the image-domain implementation are the objective function, defined using time-lag extended images obtained by wave-equation migration, and its gradient, computed using the adjoint state method. We derive the gradient calculation and compare its elements one-to-one with their data-domain counterparts. The two implementations are similar since they use the same wavefields constructed using the same wave-equation, boundary and initial conditions. The difference is that the objective functions are formulated before or after summation over experiments. The image-domain implementation evaluates the objective function throughout the model space, while the data-domain implementation evaluates the objective function only on the surface. The image-domain method constrains better the velocity model, while preserving all other important characteristics of its data-domain counterpart.