A goal of geophysical inversion is to identify all models which give an acceptable misfit between predicted and observed data. However, because of the complexity of Earth structure, the non-linearity of physical processes in the Earth, and the insufficiency of geophysical data, many geophysical inverse problems may have a large number of distinct, acceptable solutions. These problems may be characterized by a complicated surface for the misfit function in the solution parameter space. For exploring such a surface, direct inversion and simple random search methods are often inadequate. However, directed search methods such as the genetic algorithm can be configured to balance convergent and random processes to find large sets of solutions that span the acceptable regions of complicated misfit surfaces.