Bound-constrained tomography for anisotropic and Q model building

Reflection tomography is an effective way to build anisotropic velocity and Q models for seismic imaging. However, reflection tomography suffers from a nonuniqueness problem, even when it only inverts for a single parameter, such as velocity (Stork, 1992). When anisotropy exists, surface seismic data alone cannot sufficiently determine the anisotropy parameters and there is ambiguity between them. Unconstrained reflection tomography may yield undesired or even non-physical models. To better constrain the tomographic inversion, additional information has to be incorporated into the system of equations. For example, Li et al. (2014) apply the results from stochastic rock physics modelling as an additional constraint to migration velocity analysis for anisotropic parameters. Zhou et al. (2014) incorporate well data into the ray-based tomographic equation system. In many cases, especially in frontier areas, little geologic or well information is available, complicating the process of model building. However, most likely the rough bounds of the model parameters in an area are known. Such bounds can be incorporated into the equation system as additional constraints. Vollebregt (2014) proposes a bound constrained conjugate gradient algorithm for an optimization system constrained by positive solutions. In this paper, we generalize this algorithm to accommodate spatially variant lower and upper bounds for seismic model parameters and use these bounds to constrain reflection tomography.