Chapter 6
Stochastic Inverse Problems: Models
and Metrics
6.1 Introducing the Problem
Over the past 2 years, we have been developing a theory of uncertainty quantification
and propagation that is computationally feasible with large numbers of unknowns.
We have applied it to a problem of characterizing the eddy-current response of a
shot-peened surface, where the surface is modeled as a one-dimensional random
conductivity field with a known covariance function. We are currently extending
the model to more general materials characterization problems, such as modeling
two-dimensional random anisotropic grain noise in titanium alloys. In this case, we
assume the existence of a (two-dimensional) covariance function for the random
distribution of Euler angles that define the orientation of each crystallite within the
material.
With this background, we want to develop a theory of stochastic inverse problems
for more traditional eddy-current NDE flaw characterization and sizing. Instead of a
random material, we assume that the flaw can be characterized as a random process.
That this is a reasonable approach is suggested by reference to Fig. 6.1, which shows
the typical shape of fatigue-crack growth progression in cold-worked fastener holes.
Clearly, the ensemble of cracks cannot be modeled by a simple canonical shape with
three parameters, length, width, height, so we will need to invoke a stochastic model
for analyzing such cracks.
With such a stochastic model, we can draw parallels between ‘probability of
detection’ (POD) and ‘likelihood of inversion’ (LOI). In the former, we are given
a flaw, and ask ourselves, ‘Can we detect it, and what are the metrics that measure
our success?’ In the latter, we are given data, and ask ourselves, ‘Can we associate
a flaw with them, and what are the metrics that measure our success?’
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
H. A. Sabbagh et al., Advanced Electromagnetic Models for Materials
Characterization and Nondestructive Evaluation, Scientific Computation,
https://doi.org/10.1007/978-3-030-67956-9_6
143
Stochastic Inverse Problems: Models
and Metrics
6.1 Introducing the Problem
Over the past 2 years, we have been developing a theory of uncertainty quantification
and propagation that is computationally feasible with large numbers of unknowns.
We have applied it to a problem of characterizing the eddy-current response of a
shot-peened surface, where the surface is modeled as a one-dimensional random
conductivity field with a known covariance function. We are currently extending
the model to more general materials characterization problems, such as modeling
two-dimensional random anisotropic grain noise in titanium alloys. In this case, we
assume the existence of a (two-dimensional) covariance function for the random
distribution of Euler angles that define the orientation of each crystallite within the
material.
With this background, we want to develop a theory of stochastic inverse problems
for more traditional eddy-current NDE flaw characterization and sizing. Instead of a
random material, we assume that the flaw can be characterized as a random process.
That this is a reasonable approach is suggested by reference to Fig. 6.1, which shows
the typical shape of fatigue-crack growth progression in cold-worked fastener holes.
Clearly, the ensemble of cracks cannot be modeled by a simple canonical shape with
three parameters, length, width, height, so we will need to invoke a stochastic model
for analyzing such cracks.
With such a stochastic model, we can draw parallels between ‘probability of
detection’ (POD) and ‘likelihood of inversion’ (LOI). In the former, we are given
a flaw, and ask ourselves, ‘Can we detect it, and what are the metrics that measure
our success?’ In the latter, we are given data, and ask ourselves, ‘Can we associate
a flaw with them, and what are the metrics that measure our success?’
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
H. A. Sabbagh et al., Advanced Electromagnetic Models for Materials
Characterization and Nondestructive Evaluation, Scientific Computation,
https://doi.org/10.1007/978-3-030-67956-9_6
143