Logarithmic Fourier transformation

INTRODUCTION SUMMARY We present an exact and analytical expression for the Fourier transform of a function that has been sampled logarithmically. The procedure is significantly more efficient computationally than the fast Fourier transformation (FFT) for transforming functions or measured responses which decay slowly with increasing abscissa value. We illustrate the proposed method with an example from electromagnetic geophysics, where the scaling is often such that our logarithmic Fourier transform (LFT) should be applied. For the example chosen, we are able to obtain results that agree with those from an FFT to within 0.5 per cent in a time that is a factor of 10 shorter. Potential applications of our LFT in geophysics include conversion of wide-band electromagnetic frequency responses to transient responses, glacial loading and unloading, aquifer recharge problems, normal mode and earth tide studies in seismology, and impulsive shock wave modelling.