CLASH IN PHILOSOPHIES

One philosophy of geophysical data analysis called ``inverse theory'' says that missing data is irrelevant. According to this philosophy a good geophysical model only needs to fit the real data, not interpolated or extrapolated data, so why bother with interpolated or extrapolated data? It must be a hoax. Most academic inverse theorists as well as our own Peter Mora seem to belong to this school of thought. Even some experienced practitioners belong to this school of thought. My good friend Boris Zavalishin says, ``Don't trust the data you haven't paid for.''

Another philosophy is that 40 years of geophysical data processing cannot be all wrong. There must be some good reason why we interpolate and extrapolate data though perhaps we do not clearly explain why. Let me cite some examples:

• Spectral analysis with prediction-error filters shows that unexpectedly high resolution can be achieved reliably with certain kinds of data sets. Conceptually this amounts to extending the data set with nonzero values beyond the recorded region. Familiar statments about limited resolving power generally presume data vanishes beyond where it is recorded.
• By now we are all accustomed migration side-boundary artifacts. We eliminate these by either tapering data near the sides (falsifying the data) or by extending nonzero data into the region beyond where data is recorded.
• Everyone does velocity analysis by hyperbola scan whereas most people do migration by Fourier analysis or finite differences. Since they are mathematically equivalent, why is the velocity analysis never implemented by the other two methods? Maybe it has something to do with the spatial sampling of the wide-offset data.
• Certain idealized data sets can be much more effectively interpolated and extrapolated by human beings than by any known mathematical analysis. We frequently see data with a limited dip spectrum but seem unable to exploit it because of our clumsiness with spatially aliased data.

Are we wasting our time interpolating and extrapolating the observed data field, or is some good purpose being served, if so, what? How can the conflict between the ``inversion'' school of thought and the ``missing data'' school of thought be resolved?