(t),
called the "mother wavelet:"The use of wavelet transforms in geophysics is relatively recent (e.g., Foufoula-Georgiou and Kumar, 1995; Kumar and Foufoula-Georgiou, 1997). The wavelet transform represents an alternative to the classical spectral analyses, such as the fast Fourier transform (FFT) or Gabor transform (Carmona et al., 1998), for time-frequency analysis; it allows not only frequency but also time localization. Wavelet transform then enables us to analyze nonlinear and nonstationary signals that contain multiscale features such as downhole logging measurements or core petrophysical measurements.
The algorithm we applied
to our data sets is the one developed by Barthes and Mattei (1997), based on the
theory initiated by Morlet at al. (1982a, 1982b) and later formalized by
Grossman and Morlet (1984), Goupillaud et al. (1984), and Meyer (1992). The
coefficient, W, of the wavelet transform can be defined as the
convolution product of a signal, s(t), function of time, t, by
functions obtained by dilatation (or contraction) and temporal translation of a
function, (t),
called the "mother wavelet:"
(1)
where,
= the analyzing wavelet function.
The factor, 
, allows for normalization.
One drawback of this definition is that the values of the time-frequency resolution of the analyzing wavelet depend on the frequency:
(2)
where a is a parameter that allows us to tune time/frequency resolution (dilatation/contraction of the mother wavelet).
To overcome this difficulty and obtain constant resolution, whatever the frequency, both in time and frequency, Barthes and Mattei (1997) took a proportional to f -2. The wavelet transform is then redefined as
(3)
The Morlet wavelet used in this work is a locally periodic wavetrain. It is obtained by taking a complex sine wave and localizing it with a Gaussian (bell-shaped) envelope:
(x)
= e -ax2 x
e 2
ix.
(4)
We selected this wavelet because its form is similar to the signals in our data. For more information about the choice of a mother wavelet, see Torrence and Compo (1998).
The Morlet wavelet is a complex wavelet that can be decomposed into real and imaginary parts (Fig. F6). The results in the time-frequency domain can be presented in three ways: the real part, the modulus, or the phase of the wavelet coefficients. Because interpretation of the phase is more complex, we chose to represent on a time-frequency map the real part or the modulus of the results in the following discussion. In this case, a periodic signal appears as a crest (straight and horizontal contour lines), and amplitude modulation of this signal will modulate the amplitude of the crest (ellipsoidal contours).
