Philosophy of inverse problems

Physical processes are often simulated with computers in much the same way they occur in nature. The machine memory is used as a map of physical space, and time evolves in the calculation as it does in the simulated world. A nice thing about solving problems this way is that there is never any question about the uniqueness of the solution. Errors of initial data and model discretization do not tend to have a catastrophic effect. Exploration geophysicists, however, rarely solve problems of this type. Instead of having (x,z)-space in the computer memory and letting t evolve, we usually have (x,t)-space in memory and extrapolate in depth z. This is our business, taking information (data) at the earth's surface and attempting to extrapolate to information at depth. Stable time evolution in nature provides no ``existence proof'' that our extrapolation goals are reasonable, stable, or even possible.

The time-evolution problems are often called forward problems and the depth-extrapolation problems inverse problems. In a forward problem, such as one with acoustic waves, it is clear what you need and what you can get. You need the density $\rho (x,z)$ and the incompressibility K(x,z), and you need to know the initial source of disturbance. You can get the wavefield everywhere at later times but you usually only want it at the earth's surface for comparison to some data. In the inverse problem you have the waves seen at the surface, the source specification, and you would like to determine the material properties $\rho (x,z)$ and K(x,z). What has been learned from experience is that routine observations do not give reasonable estimates of images or maps of $\rho$ and K.