Adjoint-state methods (ASMs) have proven successful for calculating the gradients of the functionals commonly found in geophysical inverse problems. The 3D ASM image-domain tomography (IDT) formulation of the seismic velocity estimation problem highlights imperfections in migrated image volumes and, using appropriate penalty functions (e.g., differential semblance), forms an objective function that can be minimized using standard optimization approaches. For time-lapse (4D) seismic scenarios, we show that the 3D ASM-IDT approach can be extended to multiple (e.g., baseline and monitor) data sets and offers high-quality estimates of subsurface velocity change. We discuss two different penalty operators that lead to absolute and relative 4D inversion strategies. The absolute approach uses the difference of two independent 3D inversions to estimate a 4D model perturbation (i.e., slowness squared). The relative approach inverts for the model perturbation that optimally matches the monitor image to the baseline image — even if migrated energy is imperfectly focused. Both approaches yield good 4D slowness estimates; however, we assert that the relative approach is more robust given the ubiquitous presence of nonrepeatable 4D acquisition noise and imperfect baseline model estimates.

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