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
|
UA&UQ
28
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
(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
|
UA&UQ
28
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
