Missing-data restoration

  A brief summary of chapters and is that ``the answer'' is the solution to an inversion problem--a series of steps with many pitfalls. Practitioners often stop after the first step, while academics quibble about the convergence, i.e., the last steps. Practitioners might stop after one step to save effort, to save risk, or because the next step is not obvious. Here we study a possible second step--replacing the zero-valued data presumed by any adjoint operator with more reasonable values.

A great many processes are limited by the requirement to avoid spatial aliasing--that no wavelength should be shorter than twice the sampling interval on the data wave field. This condition forces costly expenditures in 3-D reflection data acquisition and yields a mathematical dichotomy between data processing in exploration seismology and data processing in earthquake seismology.

The simple statement of the spatial Nyquist requirement oversimplifies real life. Recently, S. Spitz (1991) showed astonishing results that seem to violate the Nyquist requirement. In fact they force us to a deeper understanding of it. In this chapter we will discuss many new opportunities that promise much lower data-acquisition costs and should also reduce the conceptual gap between exploration and earthquake seismology.