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Classification of sonic wave Chapter 9 281

5.3 Characterization of material containing static discontinuities
of various spatial distributions

5.3.1 Background
Due to the variations in geomechanical properties and stress, various types of
spatial distribution of discontinuities develop in materials. In this section, nine
classifiers (discussed in Section 4.1) process compressional wavefront travel
times to categorize materials containing discontinuities in terms of the distribu-
tions of the discontinuities. A intensity function is used to define the spatial var-
iability in the density of discontinuities for a material; following that, the
locations of 100 fractures are assigned using the nonhomogeneous Poisson pro-
cess [18, 20]. Intensity functions and the corresponding materials containing
discontinuities are shown in Fig. 9.25.
We implement acceptance-rejection method to generate the nonhomoge-
neous Poisson process for assigning the location of discontinuities, as shown
in the upper plots of Fig. 9.25. First, an intensity function λ(x,y) is defined
on the domain of investigation denoting the aerial extent of the material. The
intensity function is normalized with respect to the maximum value of the inten-
∗ ∗
sity (λ ) in the domain to obtain an acceptance probability p(x,y) ¼ λ(x,y)/λ .
Homogeneous Poisson process is used to pick a location of discontinuity in
the material, which is then accepted or rejected based on the acceptance prob-
ability p(x,y), thereby embedding discontinuities in the material based on a non-
homogeneous Poisson process. According to the intensity function (bottom
plots of Fig. 9.25), the locations of discontinuity generated by the homogeneous
Poisson process from the region with a higher intensity are more likely to be
retained.
Fig. 9.25A shows a material containing discontinuities generated using con-
stant intensity function. The distribution of discontinuities is similar to random
distribution. Fig. 9.25B shows material containing discontinuities generated
using linear intensity function λ(x,y) ¼ y. The possibility of acceptance
increases linearly along the y-axis toward the positive y-direction. The origin
of the coordinate system is located at the upper left corner of the material. Con-
sequently, the density of discontinuities is higher near the lower boundary adja-
cent to the transmitter-bearing boundary. Fig. 9.25C shows the material
containing discontinuities generated using a unimodal Gaussian function as
the intensity function:

2 2
ð ð
ð
λ x, yÞ ¼ c∗exp ð d ∗ x x 0 Þ + y y 0 Þ ÞÞ (9.9)
ð
where x 0 and y 0 are the center of the Gaussian distribution, d controls the var-
iance of the distribution, and c controls the minimum value of the intensity func-
tion. In this experiment, c is set to 1, d is set to 0.00005, and x 0 and y 0 are both set
to 250, which is the center of material. The acceptance probability is high at the
center of the material and drops away from the center of the material. Fig. 9.25D