On stability of 1D exact inverse methods

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Abstract
It is well known that despite the fact that one-dimensional inverse methods are well developed there are still difficulties when these methods are applied to real seismic data. There are several reasons why these methods usually fail. Real seismic data are corrupted by noise. The data typically do not contain the low and high frequencies. This can dramatically degrade the reconstruction of the acoustic impedance as well as affect the resolution. The stability of one-dimensional exact methods is considered. Despite the fact that all these methods require the source time function to be a Dirac delta-function, the author examines the possibility of applying these methods to an arbitrary source function. It is shown that the results of the inversion procedure are very sensitive to the characteristics of the source wavelet chosen for inversion. In practice (in exploration geophysics, for example) it is almost impossible to measure the source wavelet precisely. Therefore, it appears very important to known in which cases small errors in the source wavelet will not cause large errors in a solution. (In other words, in which cases the problem will be well-posed even though the source time function is not impulsive.) Based on the analysis presented the author draws the conclusion that small uncertainties inherent in the estimate of the source spectrum can lead to severe instabilities in the low-frequency portion of its spectrum whereas small uncertainties in the high-frequency portion of the source spectrum hardly affect the recoverability of the acoustic impedance.