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TVF: Theoretical Basis

TVF: Theoretical Basis.

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TVF: Theoretical Basis

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  1. TVF: Theoretical Basis The Time Variable Filtering (TVF) is a filtering technique that does not give a dispersion curve but a smooth signal (a signal time-variable filtered), in which all effects of noise, higher modes and other undesirable perturbations have been removed (Cara, 1973). For it, a Fourier synthesis of the observed signal is performed (Brigham, 1988), in which only are considered the Fourier harmonics in the neighboring of the group time tg(f), given by the formula: where F(f) is the Fourier spectrum of f(t), computed applying the FFT forward to the observed signal f(t). The expression for time-frequency window w(t,f) is presented in the following slide (Bath, 1974).

  2. TVF: Theoretical Basis where tw is given by The parameters a and b are constant during the filtering process (Cara, 1973). Usually, the values for these constant parameters are a = 5 and b = 0. This filtering technique requires a starting dispersion curve tg(f), to perform the Fourier synthesis. This dispersion curve can be provided by the application of the MFT to the observed signal f(t), previously to the computation of the TVF.

  3. TVF: Flow Chart Preprocessed signal f(t) (observed seismogram with instrumental correction) f(t) F(f) FFT Selection of an initial dispersion curve tg(f) = D/Ug(f) Time-variable filtering Time window w(t,f) Filtered signal g(t)

  4. TVF: An Example The above-described filtering process, as an example, has been applied to the trace shown below, which has been instrumentally corrected. The starting dispersion curve tg(f) necessary to perform the TVF, has been obtained from this observed trace by application of the MFT, as it is shown in the PPT presentation: MFT.

  5. TVF: An Example A Fourier synthesis of the observed signal shown in (a) is performed, considering only the Fourier harmonics in the neighboring of the dispersion curve shown in (b), to obtain the time-variable filtered signal shown in (c).

  6. TVF: References Bath M. (1974). Spectral Analysis in Geophysics. Elsevier, Amsterdam. Brigham E. O. (1988). The Fast Fourier Transform and Its Applications. Prentice Hall, New Jersey. Cara M. (1973). Filtering dispersed wavetrains. Geophys. J. R. astr. Soc., 33, 65-80. TVF: Web Page http://airy.ual.es/www/TVF.htm

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