Page 286 - Machine Learning for Subsurface Characterization

248 Machine learning for subsurface characterization

will increase with the increase in number of source-receiver pairs. In the real
world, the number of source-receiver pairs can be five times or higher as
compared with that implemented in this study.
c. As a proof of concept, each receiver detects only the first arrival (i.e., wave-
front travel time) of compressional wave, and the characterization is accom-
plished only with compressional travel-time data, which are very limited
information related to the wave propagation through a fractured material.
Characterization performance will increase with the use of shear wavefront
travel times or full compressional/shear waveforms.
d. Feature importance is performed using feature permutation importance to
identify the optimal placement of receivers/sensors for accomplishing the
desired static characterization of discontinuities by processing the multi-
point measurements of compressional wavefront travel times.
e. The proposed method can approximate certain statistical parameters of the
system of discontinuities, like dominant orientation and intensity of distri-
bution, with simple source-receiver configurations.

3 Fast-marching method (FMM)
3.1 Introduction

The fast-marching method (FMM) is developed to solve the eikonal equation,
expressed as
1
j ru xðÞj ¼ for x 2 Ω (9.1)
f xðÞ
u xðÞ ¼ 0 forx 2 ∂Ω (9.2)
where u(x) represents the wavefront travel time (i.e., time of first arrival) at
location x, f(x) represents the velocity function for the heterogeneous material,
Ω is a region with a well-behaved boundary, ∂Ω is the boundary, and x is a spe-
cific location in the material. Eikonal equation characterizes the evolution of a
closed surface Ω through a material with a specific velocity function f. FMM is
similar to the Dijkstra’s algorithm. Both algorithms monitor a collection of
nodes defining the region Ω and expand the collection of nodes by iteratively
including a new node just outside the boundary ∂Ω of the region Ω that can be
reached with the least travel time from the nodes on the boundary ∂Ω. FMM has
been successfully applied in modeling the evolution of wavefront in heteroge-
neous geomaterial [16]. In this study, FMM is used to simulate the compres-
sional wavefront propagation in fractured material.

3.2 Validation

FMM predictions are validated against analytical solutions and against the pre-
dictions of k-Wave (MATLAB toolbox) in materials with and without fractures.