new sensor data, improving accuracy
and reliability by addressing the limita-
tions of static approaches. To enhance
fault detection in feedback control
systems, adaptive residual generators are
applied to update data-driven models in
response to system anomalies, demon-
strating effectiveness in improving
robustness and scalability in discontinu-
ity detection [37].
UA&UQ Methodologies in NDE
Various UA&UQ methodologies have
been applied across different NDE
techniques to improve inspection reli-
ability, including probabilistic [24], sta-
tistical [38], simulation-based [39], and
AI-driven approaches [40]. This section
summarizes and explains how these
methodologies address uncertainty and
highlights their diverse applications in
industrial NDE. A summary of UA&UQ
methods is presented in Figure 1.
Probabilistic Approaches
Probabilistic UQ methods address
uncertainties by considering variabil-
ity in measurements, sensor errors, or
discontinuity properties as random
variables. By analyzing their probability
distributions, quantitative uncertainty
bounds can be established for detec-
tion reliability, discontinuity sizing, and
maintenance decisions.
PROBABILITY OF DETECTION (POD)
WITH PROBABILITY DISTRIBUTIONS
FOR UNCERTAINTY QUANTIFICATION
POD is a commonly used method of
quantifying an NDE system’s reliability.
It estimates the probability of detecting a
discontinuity of a given size and enables
informed decisions about the disconti-
nuity’s severity and its impact on struc-
tural integrity. POD analysis is frequently
categorized into two types: signal
response (â vs. a) and hit/miss, each
using distinct data types and analysis
approaches [111–113]. Signal response
methods rely on continuous data, while
hit/miss methods use binary outcomes,
both requiring accurate probability
models for quality POD estimation [41].
Hit/miss analysis has been widely
applied in quantitative visual assess-
ments of discontinuity response for
system reliability evaluation [42], includ-
ing applications such as visual inspec-
tion [43], magnetic particle inspection
[44], and ultrasonic testing [45]. The
â vs. a approach is applicable when a
quantitative signal response is available
and found to be correlated with discon-
tinuity size, as is typically attainable with
techniques like ultrasonic [46] or eddy
current inspection [47].
In an â vs. a POD study, the inspec-
tion produces a continuous response
estimate, denoted ​ Â​ as a function of the
true discontinuity size, ​ ​ Responses at
or above the decision threshold ​​ ​ dec​​​ are
classified as detections. The probability
of detection for discontinuity size ​ ​ is
expressed as: ​ ​ ​
(​​1)​​ ​ POD(a)​ ​ =P​(Â ≥ ​ Âdec​​ ​ a)​ =
1 − ​ F​{Â ​ |a}​​​​ (​Â​dec​​)​​​
where ​​ ​ ​ {​​Â ​ |​​ a}​​​​​ ​ (Â​dec​​)​​ ​ is the cumula-
tive distribution function (CDF) of the
response ​ ​ conditioned on discontinuity
size ​ a​ It represents the probability that
the response falls below the threshold
subtracting this value from 1 yields the
detection probability.
The black line in Figure 2 shows an
example of a mean ​​ OD(​​a)​​​​ ​ ​ curve with
respect to discontinuity size ​ ​ [14]. For
small discontinuities with low response
relative to the NDE detection limit, the ​​
POD(​​a)​​​​ ​ ​ approaches zero, increasing
toward 1 as discontinuity size increases
with a response well beyond the detec-
tion limit.
Due to uncertainties and limited
samples from real-world NDE inspec-
tions, the POD curve is typically eval-
uated with a focus on the one-sided
(upper) confidence bound, shown as
the solid blue line. The metric ​​ ​ 90/95​​​ is
widely used as the measure of detect-
ability for NDT applications and rep-
resents the discontinuity size for which
there is at least 90% detection probability
with 95% confidence [48].
For a hit/miss study, each inspection
of a discontinuity of size ​ a​ either detects
Probabilistic methods
• Probability distributions
• Reliability analysis
Simulation-based
methods
• Monte Carlo simulations
• Bayesian inference
• Polynomial chaos expansion
AI-driven methods
• Bayesian neural networks
• Monte Carlo dropout
• Deep ensembles
Statistical methods
• GUM-based measurement-
uncertainty analysis
• Confidence intervals
Popular UQ
methods in NDE
applications
Figure 1. Popular uncertainty analysis (UA) and uncertainty quality (UQ)
methods in NDE applications.
POD (a)
POD curve
100%
90%
95% confidence bound
a90/95 Crack length (a)
Figure 2. Typical probability-of-detection (POD) curve [14].
A U G U S T 2 0 2 5 • M AT E R I A L S E V A L U AT I O N 27

the discontinuity (hit, ​D =1​ or does not
(miss, ​ =0​ The probability of detection
is the conditional probability: ​ ​
(2)​ POD(a)​ ​ =P​(D =1 |a)​​
Probability distributions such as
Normal (Gaussian), Log-Normal, and
Beta distributions are commonly used to
build POD curves, providing a statistical
foundation for addressing uncertainty.
The selection of appropriate probability
models is crucial for deriving meaningful
POD curves.
Ñ The Normal (Gaussian) distribution
models symmetrical variations around
a mean value, making it well suited
for cases where errors result from
random deviations that are evenly
distributed, such as sensor noise
and measurement fluctuations. It is
frequently applied in ultrasonic testing
or radiographic inspection, where
measurement noise and variability are
common [49].
Ñ The Log-Normal distribution is
effective for modeling asymmetric
uncertainties, where smaller values
occur frequently but large deviations
are possible. Its multiplicative nature
aligns with processes like corrosion
propagation or material wear [50].
Ñ The Beta distribution is commonly
used when the variables are naturally
constrained, where it quantifies
detection confidence levels by
modeling bounded uncertainties
(between 0 and 1). It has proven
effective for representing disconti-
nuity detection probabilities, given its
bounded nature aligns well with the
physical constraints often encountered
in NDE inspections, where measure-
ments or probabilities cannot exceed
realistic limits [51].
RELIABILITY-BASED METHODS FOR
DISCONTINUITY DETECTION
Reliability-based UQ methods evaluate
the probability of discontinuity detection
failure by incorporating probabilistic
models and statistical reliability analysis.
Instead of merely identifying a disconti-
nuity, reliability-based UQ predicts the
likelihood of a detected discontinuity
leading to failure under various operat-
ing conditions, making these methods
essential in industries with strict safety
requirements.
First-order and second-order reli-
ability methods (FORM and SORM) are
typical reliability approaches that lin-
earize the boundary between safe and
failure conditions. Instead of considering
every possible scenario, FORM locates
the most probable point (MPP) on the
limit-state surface, ​​ ​ (​​X)​​ ​ =0​​ by trans-
forming the original variables into stan-
dard-normal U-space and then lineariz-
ing ​ ​ at that point via a first-order Taylor
expansion. Central to this method is the
Hasofer–Lind reliability index, ​ ​ which
quantifies the shortest distance from
the origin to the limit-state surface in
standard normal space [52]. The failure
probability is approximated by: ​ ​
(3)​ ​ Pf​​​ ≈ ϕ(−β)​​ ​
where ​
ϕ​ is the standard-normal CDF.
SORM improves on FORM by
adding the second-order terms of the
Taylor series, fitting a curved surface
at the MPP [53]. It corrects FORM’s
linear approximation with curvature
factors derived from the principal radii
of the limit-state surface, yielding more
accurate failure probabilities for highly
nonlinear problems, which are shown in
Figure 3.
Recent innovations have enhanced
FORM’s capabilities by combining it
with complementary techniques. For
instance, Zhu and Xiang [54] paired
FORM with the stochastic pseudo-
excitation method (SPEM) to improve
dynamic reliability analysis under
random excitation. Such hybrid
approaches address FORM’s limitations
in handling complex system behaviors,
expanding its applicability in NDE.
Monte Carlo reliability analysis
(MCRA) is another prominent reli-
ability method for UQ. It operates on
the principle of random sampling to
estimate failure probabilities, making
it particularly effective for complex or
high-dimensional problems where tradi-
tional methods like FORM may struggle.
The strength of MCRA lies in its ability
to handle nonlinear and discontinuous
performance functions without relying
on gradient-based approximations
[55]. Its versatility extends to various
NDE contexts, including structural reli-
ability analysis [56] and evaluation of
cyber-physical systems [57], where it can
simulate complex failure modes and
operational constraints.
Current research gaps include
the need for improved integration of
FORM and Monte Carlo methods and
the development of adaptive sampling
strategies to enhance computational effi-
ciency. Machine learning advancements
could further refine reliability-based
UQ in NDE by optimizing discontinu-
ity detection under uncertainty [58].
Addressing these challenges will advance
the reliability and accuracy of NDE tech-
niques for complex engineering systems.
Statistical Approaches
Statistical UQ methods provide probabi-
listic evaluations of detection accuracy,
discontinuity size estimation, and mea-
surement noise effects. By integrating
statistical techniques such as measure-
ment uncertainty analysis, confidence
intervals, and resampling methods,
engineers can enhance sensor calibra-
tion, discontinuity characterization,
and decision-making in various NDE
applications.
U
2
U1
g 0
o
β
FORM
MPP u*
SORM
g 0
g =0
Figure 3. Comparison of first-order and
second-order reliability methods (FORM
and SORM) [53].
NDT TUTORIAL
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UA&UQ
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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