A comprehensive solution to the linear deconvolution problem

TL;DR: In this article, the authors used the method of Backus and Gilbert to generate localized averages of the model which are unique except for a statistical uncertainty caused by errors in the data.

Abstract: Summary Linear inverse theory is used to deconvolve a data set when the blurring function or source wavelet is (approximately) known. Rather than attempting to find one of infinitely many models which fits the data this paper uses the methods of Backus and Gilbert to generate localized averages of the model which are unique except for a statistical uncertainty caused by errors in the data. The averages, their statistical error and the associated averaging window completely codify our knowledge about the model. Averages with lower standard deviations can be had by sacrificing resolution and the investigator is free to choose those results which are most meaningful. Moreover, this method is optimum in the sense that no other averaging window can be constructed which has greater resolving power and yet produces averages with the same statistical accuracy. Our deconvolution method in the time domain is shown to be very similar to finding the inverse filter of the source wavelet, and indeed the averaging window is simply the convolution of these two functions. However, by investigating the trade-off between resolution and accuracy we have shown that the data errors can be much more important than the parameters of the Wiener optimum inverse filter such as the length of that filter and the desired location of the output spike. In the frequency domain the equations for trading off accuracy and resolution have been developed and the computations are seen particularly to be simple because no matrix inversion is required. Sufficient examples will be presented to show the importance of incorporating the observational errors into the deconvolution procedure. Additionally, we have shown how to reduce the sidelobes of the averaging windows by shaping them into Gaussian functions of a predetermined width, have looked at the effects of using a zero area source function characteristic of seismic problems, and have attempted a deconvolution when the wavelet was only approximately known. The frequency domain deconvolution filter derived here is also compared quantitatively with those derived intuitively by Helmberger and Wiggins and Deregoiwski. Lastly, we show how information in the averages and averaging windows can be used to construct a parametric model, composed of a series of delta functions, which fits the data. Such a model is of importance in seismological and spectroscopic studies.