Distribution of sparsity

We rarely have data in which the transition from good coverage to poor is as abrupt as here, but we generally have data that terminates even more abruptly at the edges of a survey. There is also the possibility of coverage that is uniformly sparse, so that we would have no good region in which to learn the data statistics. In principle we should be simultaneously estimating the solution and its multidimensional prediction-error filter. Such simultaneous estimation is nonlinear. Thus there are well-known dangers, but the problem itself is not ill-conceived or impossible to approach, as the one-dimensional example ofFGigure  shows. Although real problems are nonlinear, it is often realistic to approach them textbook-style, as a sequence of linearized approximations. Sometimes ingenious tricks can be brought to bear, as in Figure .

Current practice in the seismic-exploration industry meets the requirements perfectly of a nonlinear problem that is near to linear, because the high costs of occupying many data stations limit the surveys to avoid aliasing at central frequencies, while allowing aliasing at the highest frequencies (which define the resolution).