Subspace methods for large inverse problems with multiple parameter classes

Most nonlinear inverse problems can be cast into the form of determining the minimum of a misfit functional of model parameters. This functional determines the misfit between observations and the corresponding theoretical predictions, subject to some regularization conditions on the form of the model. When there is only one type of parameter in the model, methods based on gradient techniques work well, especially when information on rate of change of gradients is included. In the case of problems depending on multiparameter classes, simple gradient methods mix parameters of different character and physical dimensionality. This may lead to rather poor convergence and strong dependence on the scaling of the different parameter types. These difficulties can be overcome by replacing a gradient step by a local minimization in a subspace spanned by a limited number of vectors in model space. The basis vectors for the subspace should be chosen in the directions determined by the variation of the misfit functional with respect to each of the parameter types, with supplementation if required by additional vectors representing the rate of change of the gradient partitions. The construction of the perturbation requires the inversion of a matrix with the dimensions of the subspace which is easily accomplished. Such a subspace scheme takes into account the different functional dependences on the various parameter types in a balanced way. The update to the current model does not depend on the scaling of the individual parameter classes. The subspace method is flexible and can be adapted to a wide range of choices of misfit criterion and modes of representation of the parameter classes. This style of iterative subspace procedure is well adapted to nonlinear problems with dependence on many parameters and can be successfully applied in a variety of problems, e.g. seismic reflection tomography, the simultaneous nonlinear determination of earthquake locations and velocity fields and in the inversion of full seismic waveforms.