Page 65 - Machine Learning for Subsurface Characterization

Characterization of fracture-induced geomechanical alterations Chapter 2 51

l Number of signal zero crossings: A zero-crossing is a point where sign of
mathematical function changes, and it is represented by the intercept of
function. This feature represents the number of times the value of signal
is zero in a time interval representing the oscillatory nature and
wavelength of the signal.
A signal is considered stationary when its statistical measures do not change
over time, for example, rotating machinery. Nonstationary signals, on the
other hand, have time-varying statistical features as the frequency spectrum
of such signals change over time, for example, the speech signal and seismic
signal. The features outlined earlier (those used for generating the first set of
features) are most useful in detecting changes in the “form” of a time-
invariant signals. They are used extensively in machine learning assisted
damage detection methods in ball bearings and rotating electrical machinery
[14] that produce stationary signals. Seismic signals and ultrasonic shear
waveforms, on the other hand, are examples of nonstationary signals and
feature engineering methods suited for stationary signals are not designed to
capture the time-varying parameters in nonstationary signals [15], other than
breaking the signals into time windows (segments) and the deriving the
stationary statistical features for each segment.
Few popular feature engineering methods for nonstationary signals are short-
time Fourier transform (STFT), the continuous wavelet transform (CWT),
and the empirical mode decomposition (EMD). Fourier transform (FT)
decomposes a signal into a sum of sinusoids. Unlike the widely implemented
Fourier transform, the CWT decomposes the signal into shifted and stretched
(or compressed) variations of a wavelet. A wavelet is a wavelike oscillation
that is localized in nature [16]. Compressing or stretching a function is termed
as scaling. The result of a CWT implementation is not a time-frequency map
but a map that is termed scalogram. CWT produces a time versus scale
variation and not a true time versus frequency variation [17]. In present study,
it was observed that the CWT-derived scalogram could not clearly detect the
differences between pre- and postfracture waveforms. Nonetheless, it has been
applied to analyze seismic attributes [18], vehicle-generated noise analysis
[19], and acoustic-emission studies in composite materials [20]. Another
feature engineering method for nonstationary signal is empirical mode
decomposition (EMD), which decomposes a signal into functions comprising
the intrinsic node functions (IMFs). EMD is a completely data-driven
algorithm with no predefined decomposition basis functions [21] (such as
sinusoids in FT and wavelets in CWT). IMFs are extracted recursively from
the signal. The algorithm is based on the identification of local maxima and
minima, which are used to define the upper and lower envelopes by fitting a
spline curve. The mean envelope is then subtracted from the original signal,
and the aforementioned process is repeated on the residual signal. The process
stops when the mean envelope is close to zero in the entire time series. The