For the CI check, if the confidence
interval ​ ​
_
​ ±​ half-width lies entirely
beyond the critical limit ​​ ​ crit​​​ the
discontinuity is deemed too large
and is flagged. For the HT check,
compare the calculated ​ ​ statistic
with the critical value ​​ ​ n−1, α​​​ (or use
the P-value). Reject ​​ ​ 0​​​ if ​ ​ is larger
otherwise, fail to reject ​​ ​ 0​​​ .
BOOTSTRAP RESAMPLING FOR
UNCERTAINTY ESTIMATION
Bootstrap resampling is a nonparamet-
ric statistical method used to estimate
uncertainty by resampling measured
data to construct empirical confidence
intervals. Unlike parametric methods
that assume a specific probability
distribution, bootstrap resampling is
effective for non-Gaussian data, small
sample sizes, and variable discon-
tinuity characteristics. By randomly
sampling with replacement and recal-
culating discontinuity parameters
across multiple datasets, this method
quantifies uncertainty without requir-
ing predefined data assumptions. Its
effectiveness is demonstrated in NDE
applications such as image registra-
tion uncertainty estimation, where it
outperforms traditional approaches
like the Cramér–Rao bound [64], and
in metrology, where it reduces bias in
small-sample scenarios [65].
The basic process can be described
as follows:
Ñ Collect baseline data. Obtain an
original sample of ​ ​ inspection results ​ ​
x​1​​, ​ x​2​​, …​x​n​​​ (e.g., signal amplitudes,
discontinuity sizes, POD hit/miss
data).
Ñ Resample with replacement.
Generate ​ ​ bootstrap samples. For
each ​ =1,…, B​ draw ​ ​ observations
from the baseline data with replace-
ment. Then, compute the statistic of
interest ​​ ​ ​ *​(​​b​)​​​​ (e.g., mean discontinuity
depth, POD at size ​ ​
Ñ Build the empirical distribution.
The set {​​​ ​ ​ 1​ …​T​​B}​​​​ ​ ​ approximates the
sampling distribution of the statistic
without assuming any parametric
model.
Ñ Estimate bias and standard error: ​ ​
(7)​ Bias =​
_
​ − T, SE =​ √ _______________​
1 _
B − 1 ​ ​
∑
b=1​
B ​ ​ ​ (T​​b​ ​ − ​
_
​ )​​​2​​​​
where ​
T​ is the statistic from the original data,
and ​ ​ _
T​​ is the bootstrap mean.
Ñ Form confidence intervals (CIs).
Using the percentile method, define
the confidence interval using the ​
α /2​ and ​​ − ​ (​​α/2)​​​​ ​ quantiles of the
bootstrap distribution, such as [​​ ​α/2​, ​ ​
T​​1−​(​​α/2​)​​​​ The best estimate and its
uncertainty can be expressed as: ​ ​
(8)​ T =​
_
​ ± half-width​
where the half-width is half the distance
between the lower and upper limits of
the selected bootstrap interval, repre-
senting 95% CI.
Simulation-Based UQ Methods
in NDE
Another widely used approach for ana-
lyzing UQ in NDE is simulation-based
UQ methods, which allow engineers to
model the physical interactions between
inspection techniques and inspected
materials. Unlike purely statistical
methods, simulation-based UQ propa-
gates input variability through a numeri-
cal model of the inspection therefore, the
full distribution of the NDE output can be
evaluated without closed-form formulas.
Three widely used approaches in
NDE are Monte Carlo simulation (MCS)
[66], Bayesian inference via Markov
chain Monte Carlo (MCMC) [67], and
polynomial chaos expansion (PCE)
[68]. Those methods share the same
core idea: begin by assigning prob-
ability distributions to all uncertain
inputs (e.g., material properties, dis-
continuity geometry, sensor settings)
and then propagate those input uncer-
tainties through a forward NDE model.
Whether by direct random sampling
(MCS), posterior sampling (Bayesian
MCMC), or spectral projection (PCE),
each method generates multiple sim-
ulated responses, which are then used
to calculate summary statistics such as
the average outcome, its variance, and
uncertainty bounds. In other words, they
all transform uncertainties in inputs into
quantified uncertainties in inspection
responses. The comparative process is
presented in Figure 4.
MONTE CARLO SIMULATION FOR NDE
MCS is a probabilistic method that helps
estimate failure probabilities and detec-
tion uncertainties by simulating a wide
range of inspection scenarios under dif-
ferent conditions. Instead of relying on
a single fixed outcome, MCS relies on
repeated random sampling to approx-
imate the probability distributions of
output quantities, making it particularly
useful for complex systems where ana-
lytical solutions are intractable.
Key advantages are its flexibility
in handling nonlinear and correlated
inputs and its ease of implementation
[66], making it suitable for a wide range
of NDE applications. In addition, all
the independent samples run concur-
rently, making the method well suited to
modern multicore and GPU hardware.
For instance, MCS has been applied
to estimate discontinuity detection
S2: Simulation
MCS
Direct random
sampling
PCE
Surrogate
model
MCMC
Posterior
distribution
S3: Response
Mean response
95% confidence interval
S1: Input signal
Figure 4. Illustrative process of three simulation-based UQ methods: Monte Carlo simulation
(MCS), Bayesian inference via Markov chain Monte Carlo (MCMC), and polynomial chaos
expansion (PCE).
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M AT E R I A L S E V A L U AT I O N • A U G U S T 2 0 2 5

failure probability by randomly sampling
crack size, toughness, pressure, and
geometry, and then running each set
through the failure assessment diagram
(FAD) model to calculate how many
cases fail [69]. Despite it being easy to
apply, MCS still has limitations, includ-
ing slow convergence rates and sensi-
tivity to input distributions. Therefore,
careful input modeling is needed to
avoid biased results [66]. Future develop-
ments may focus on hybrid approaches,
combining Monte Carlo with other UQ
methods, such as polynomial chaos
expansion or Bayesian inference, to
further optimize performance.
BAYESIAN INFERENCE VIA MCMC
FOR ADAPTIVE NDE
Unlike traditional statistical methods,
which assume fixed probabilities,
Bayesian inference continuously refines
its predictions by incorporating prior
knowledge with real-time sensor mea-
surements. This approach is particularly
valuable for real-time monitoring and
adaptive inspection strategies, where
uncertainty evolves as new data is col-
lected. The core idea behind Bayesian
inference is to obtain a posterior distri-
bution that blends prior knowledge and
new evidence, which can be expressed as ​​
P(​​θ|​​​data)​​​​ ​ ​ where ​ ​ represents the uncer-
tain parameters. Typically, Markov chain
Monte Carlo (MCMC), Hamiltonian
Monte Carlo, or sequential Monte Carlo
are applied to sample the posterior
distribution. For each posterior sample of ​
θ​ a forward model is run to obtain dis-
tributions of inspection outputs (such as
POD, sizing errors, or failure probability).
In practice, Bayesian inference in
NDE often employs MCMC methods to
sample posterior distributions, which
are particularly effective for quantifying
uncertainties in corrosion models and
structural parameters, even with sparse
or heterogeneous data [70]. The adapt-
ability of Bayesian methods is further
highlighted by Nabiyan et al. [71], who
dynamically estimated prediction error
statistics during model updating, making
the approach suitable for handling noisy
or incomplete data.
POLYNOMIAL CHAOS EXPANSION
FOR EFFICIENT UQ
PCE is a computationally efficient method
to model uncertainty propagation in NDE
systems. Unlike MCS, PCE uses poly-
nomial expansions to approximate how
discontinuity geometry, sensor noise,
and material variability affect inspection
results. This approach reduces compu-
tational costs while maintaining high
accuracy. The process involves choosing
a polynomial basis from an orthogonal
family (e.g., Hermite, Legendre, Laguerre,
Jacobi) and truncating the order to
generate training points, using sparse
quadrature, Latin hypercube, or Sobol
sequences (often tens to hundreds of
samples, not thousands). Then, for each
training point, NDE-related simulations
are run—such as ultrasonic, eddy
current, or thermography simulations—
to compute PCE coefficients.
Applications of PCE in NDE and
related fields highlight its versatility.
For instance, Chen et al. [72] showed its
effectiveness in quantifying uncertainties
in composite cylindrical shells, address-
ing both geometric and material variabil-
ity. Similarly, its adaptability in sub-THz
antenna design and multiscale resilience
analysis has been demonstrated [73, 74]. By
combining computational efficiency with
high fidelity, PCE emerges as a powerful
tool for UQ in NDE, enabling reliable dis-
continuity detection and material charac-
terization under uncertainty.
AI-Driven UQ Methods for NDE
Artificial intelligence (AI) has revo-
lutionized discontinuity detection,
classification, and localization by
developing deep learning models for
NDE. However, traditional AI-based
approaches often operate as black
boxes, meaning they lack the ability to
quantify prediction uncertainty, which
increases the risk of false detections or
overlooked discontinuities. AI-driven
UQ methodologies have the potential to
address this issue by integrating prob-
abilistic learning frameworks. Three
widely used approaches in NDE are
Bayesian neural networks (BNNs) [75],
Monte Carlo dropout (MC dropout) [76],
and ensemble learning [77], which are
compared in Figure 5. These techniques
Uncertainty modeled via
stochastic inference
MC dropout
Input:
Output:
Uncertainty modeled through
weight distributions
BNN
Input:
Weights:
(with distribution)
Hidden layer
Weights:
(with distribution)
Output:
Ensemble learning
Uncertainty from prediction
variance across models
Neural network #1 Neural network #2 Neural network #3
Input
Output
Ensemble
Figure 5. Illustrative process of three AI-driven UQ methods: (a) Bayesian neural network (BNN) (b) Monte Carlo (MC) dropout (c) ensemble learning.
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