Flame out

The simple crosstalk problem illustrates many of the features of general modeling and inversion (finding models that fit data). We have learned the importance of weighting functions--not just their amplitudes, but also their spectral amplitudes. Certainly we have known for centuries, from the time of Gauss (see Strang, 1986), that the ``proper'' weighting function is the ``inverse covariance matrix" of the noise (a generalized relative error, that is, involving the relative amplitudes and relative spectra), formally defined in chapter . I do not know that anyone disagrees with Gauss's conclusion, but in real life, it is often ignored. It is hard to find the covariance matrix: we set out to measure a mere scalar ($\alpha$), and Gauss tells us we need to figure out a matrix first! It is not surprising that our illustrious statisticians and geophysical theoreticians often leave this stone unturned. As we have seen, different weighting functions can yield widely different answers. Any inverse theory that does not tell us how to choose weighting functions is incomplete.

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