Spectral analysis of time domain induced polarization waveforms
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Spectral induced polarization is a well-established geophysical method that can be measured in both time- and frequency domain. Time- and frequency domain methods are theoretically equivalent via the Fourier transform. However, data acquisition parameters such as the sampling frequency and pulse length affect the actual spectral content of real time domain waveforms. Comparative studies showing the practical equivalence of time- and frequency domain measurements are few and have been carried out using various approaches. Some of these studies concluded that it is not possible to resolve wide band spectral information with time domain waveforms. The few studies available presenting actual spectral analysis of time domain signals were carried out several decades ago on field data with limited data acquisition- and/or analysis methods. There is, therefore, a need to re-evaluate the possible spectral range of time domain induced polarization data considering modern instrument development during the last decades. In addition, there is a need to verify the physical importance of factors such as spectral energy distribution and measurement noise, which previously have been assumed to inherently limit the spectral content of time domain waveforms. In this study, the discrete Fourier transform is applied to analyze the spectral content of synthetic time domain waveforms with different data acquisition parameters and noise. The results are presented as the frequency domain parameters amplitude and phase and compared to the corresponding frequency domain Debye model. To study the effect of noise, a spectral analysis of synthetic full waveform data contaminated with Gaussian noise is also provided, where the noise level were guided by an example of measured field data. Finally, laboratory TD IP measurements were performed on a test circuit, transformed to the frequency domain and compared to the theoretical response of the circuit in terms of the FD IP parameters amplitude and phase. The results illustrate that spectral information contained in the time domain waveform is, as expected, limited by the fundamental frequency of the square wave at low frequencies and the sampling frequency at high frequencies. Variation of the relaxation time shows that the ability to correctly resolve high frequency relaxations is also dependent on the sampling frequency. Furthermore, the result show that high frequency noise might distort the frequency domain representation. However, this noise can be filtered away prior to the spectral analysis, which significantly improves the resolution in frequency domain. The results of this study is in contradiction to earlier studies stating that the spectral range of time domain waveforms is inherently narrow due to the domination of the fundamental frequency over the higher harmonics. The main reason is that the relative spectral energy of different harmonics is of no importance in time domain measurements; in contrast to frequency domain measurements, the total polarization of all length scales are excited simultaneously. The results and discussion in this study are expected to contribute to an improved comprehension of the spectral range that can be achieved with time domain induced polarization data collected with modern instruments.