Calculation of Fourier Transforms by the Backus-Gilbert Method

TLDR: In this article, the linear inverse theory of Backus & Gilbert has been applied to the problem of calculating the Fourier transform of digitized data with the objective of assessing the effects of missingportions of the data series and of contamination of the signal by noise.

Abstract: Summary The linear inverse theory of Backus & Gilbert has been applied to the problem of calculating the Fourier transform of digitized data with the objective of assessing the effects of missingportions of the data series and of contamination of the signal by ' noise '. When ' noise ' in the data is of concern this method achieves a maximum decrease in the variance of the Fourier transform estimate for a minimum sacrifice in resolution, thereby optimizing the trade-off between resolution and accuracy. The effects of data gaps are easily treated and it is shown that it may sometimes be desirable to interpolate these gaps even though a large variance must be ascribed to the fabricated data. We also apply the Backus-Gilbert technique to the calculation of the reverse Fourier transform, and an application to the downward continuation of potential field data is given.