Equation 7 shows that μpod ˆ and ˆ σpod are functions of the
parameters β0 β1 and σε The Delta method [19] provides a
way to estimate these variances. For a differentiable function
with derivative (T) where [y] =µ:
(13) Var
[ g(y)
]
≈ ∑
i=1
k gi(T)(μ)2
Var[yi] 2∑gi(T)gj(T)
i j
Cov[yi,yj]
In this application, is the number of parameters, so here
k =3 Equation 14 illustrates how the method can be used to
estimate Var[ˆ μpod
(14) po =Var [− ( ˆ0 − ydec) _
ˆ1
] ≈ ∂ (ˆd) po
∂ (ˆ0)
Var[ˆ0] + ∂ (ˆd) po
∂ (ˆ1)
Var[ˆ 1 ] Var[ˆd]
+ ∂ (ˆd) po
∂ (ˆε)
Var[ˆε] +2 ∂ (ˆd) po
∂ (ˆ0)
∂ (ˆd) po
∂ (ˆ1)
Cov[ˆ 0 ,ˆ1]
+2 ∂ (ˆd) po
∂ (ˆ0)
∂ (ˆd) po
∂ (ˆε)
Cov[ˆ0,ˆε] 2 ∂ (ˆd) po
∂ (ˆ1)
∂ (ˆd)op
∂ (ˆε)
Cov[ˆ1,ˆε]
=− (ˆ1)
1 Var[ˆ0] + (ˆ0 − ydec) _
ˆ1 2
Var[ˆ1] +0
− 2 (ˆ1)
1
(ˆ0 − ydec) _
ˆ1 2
Cov[ˆ 0 ,ˆ1] +0 +0
= (ˆ
1
1
2
Var[ˆ0] + 1
ˆ1 2
ˆd2Var[ˆ1] po )
+ 2
ˆ1 2
ˆdCov[ˆ0,ˆ po 1 ]
The derivatives (T) in Equation 13 can be written in a com-
plementary matrix format, The equations for μpod ˆ and ˆ σpod
(Equation 6) are needed to create the transition matrix, in
Equation 15, which contains the partial derivatives with respect
to each parameter [1–4].
Then, the matrices are useful in calculating the variance of
the POD curve, pod (ˆ μpod σpod) ˆ = UTV(ˆ)U For a simple
linear model, the resulting matrix provides the terms shown in
Equation 16.
(16) Var[ˆd] po = 1
ˆ1 2 (
Var[ˆ0] +2ˆdCov[ˆ0,ˆ po 1 ] +ˆd2Var[ˆ1] po
)
Var[ˆd] po = 1
ˆ1 2 (
Var[ˆε] − 2ˆdCov[ˆ1,ˆε] po +ˆd2Var[ˆ1] po
)
po po ˆ1 2
ˆdCov[ˆ ( po 0 ,ˆ1] − Cov[ˆ 0 , ˆε] Cov[ˆd,ˆd]=1
− ˆdCov[ˆ1,ˆε] po +ˆdˆdVar[ˆ po po 1 ])
If σε is assumed to be independent of β0 and β1 then
Cov[ˆ,ˆ] 0 ε =0 and ov[ˆ,ˆ] 1 ε =0.
The POD curve assumes a standard normal distribution,
so the Wald confidence interval is appropriate. Using a Wald
confidence interval, the probability of detecting a disconti-
nuity with % probability and % confidence is provided by
ap/q in Equation 18. Note that ap the standard deviation
at the probability associated with discontinuity size p (see
Equation 17).
(17) σap = √
_____________________________________________
Var(ˆd) po +2 zpCov(ˆd,ˆd) po po + zpVar(ˆd)op
(18) ap/q = ap + zqˆp a (ˆd po + zpˆd) po + zqσap
Equation 19 gives 90/95 which is defined as the disconti-
nuity size with 90% POD and 95% confidence:
(19) a90/95 =ˆd po + z0.9ˆd po + z0.95σa90
When performing the conversion from a linear model to a
probability curve, the variance of the POD curve is calculated
as pod (ˆ μpod σpod) ˆ = UTV(ˆ)U where is the variance-
covariance matrix from the linear model fit, and is a matrix
of derivatives of the new pod and pod with respect to each
estimated parameter. Extending this result for a model that
estimates parameters—say, 0 , β1,…,βp−2,σε2—the matrix
has a size of × p and the matrix has a size of × p so pod
will always have a final size of × 2.
3. Methods: Alternatives to the Simple Linear
Model Setup
In step 1 of the POD process, a linear model is fit to the data,
usually the simple model shown in Equation 2. However, this
model only describes the effect that discontinuity size ( has
on the signal response ( Often, as confirmed by analysis of
variance (ANOVA), the NDE response varies with multiple
variables, not just discontinuity size, and these additional vari-
ables should be included in the linear model. For instance,
the NDE response may depend on the area of a discontinuity
rather than just its length, which may require including 2 as
a variable in the linear model. Equation 5 applies to any linear
model y however, Equation 6 becomes more complex as more
variables are added, and all steps for calculating the POD curve
will require adjustment.
3.1. Variable Additions to the Simple Linear
Model Setup
The methodology defined in MIL-HDBK-1823A [4] does not
specify how to calculate POD for models beyond the simple
linear relationship between and but additional terms
are often needed. This section considers the complication of
moving from the simple linear model, = β0 + β1x to a model
with additional terms and interactions.
For example, the categorical variable of material type may
have a statistically significant relationship to the NDE response.
In that case, the indicator variable (Equation 21) may be
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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
parameters β0 β1 and σε The Delta method [19] provides a
way to estimate these variances. For a differentiable function
with derivative (T) where [y] =µ:
(13) Var
[ g(y)
]
≈ ∑
i=1
k gi(T)(μ)2
Var[yi] 2∑gi(T)gj(T)
i j
Cov[yi,yj]
In this application, is the number of parameters, so here
k =3 Equation 14 illustrates how the method can be used to
estimate Var[ˆ μpod
(14) po =Var [− ( ˆ0 − ydec) _
ˆ1
] ≈ ∂ (ˆd) po
∂ (ˆ0)
Var[ˆ0] + ∂ (ˆd) po
∂ (ˆ1)
Var[ˆ 1 ] Var[ˆd]
+ ∂ (ˆd) po
∂ (ˆε)
Var[ˆε] +2 ∂ (ˆd) po
∂ (ˆ0)
∂ (ˆd) po
∂ (ˆ1)
Cov[ˆ 0 ,ˆ1]
+2 ∂ (ˆd) po
∂ (ˆ0)
∂ (ˆd) po
∂ (ˆε)
Cov[ˆ0,ˆε] 2 ∂ (ˆd) po
∂ (ˆ1)
∂ (ˆd)op
∂ (ˆε)
Cov[ˆ1,ˆε]
=− (ˆ1)
1 Var[ˆ0] + (ˆ0 − ydec) _
ˆ1 2
Var[ˆ1] +0
− 2 (ˆ1)
1
(ˆ0 − ydec) _
ˆ1 2
Cov[ˆ 0 ,ˆ1] +0 +0
= (ˆ
1
1
2
Var[ˆ0] + 1
ˆ1 2
ˆd2Var[ˆ1] po )
+ 2
ˆ1 2
ˆdCov[ˆ0,ˆ po 1 ]
The derivatives (T) in Equation 13 can be written in a com-
plementary matrix format, The equations for μpod ˆ and ˆ σpod
(Equation 6) are needed to create the transition matrix, in
Equation 15, which contains the partial derivatives with respect
to each parameter [1–4].
Then, the matrices are useful in calculating the variance of
the POD curve, pod (ˆ μpod σpod) ˆ = UTV(ˆ)U For a simple
linear model, the resulting matrix provides the terms shown in
Equation 16.
(16) Var[ˆd] po = 1
ˆ1 2 (
Var[ˆ0] +2ˆdCov[ˆ0,ˆ po 1 ] +ˆd2Var[ˆ1] po
)
Var[ˆd] po = 1
ˆ1 2 (
Var[ˆε] − 2ˆdCov[ˆ1,ˆε] po +ˆd2Var[ˆ1] po
)
po po ˆ1 2
ˆdCov[ˆ ( po 0 ,ˆ1] − Cov[ˆ 0 , ˆε] Cov[ˆd,ˆd]=1
− ˆdCov[ˆ1,ˆε] po +ˆdˆdVar[ˆ po po 1 ])
If σε is assumed to be independent of β0 and β1 then
Cov[ˆ,ˆ] 0 ε =0 and ov[ˆ,ˆ] 1 ε =0.
The POD curve assumes a standard normal distribution,
so the Wald confidence interval is appropriate. Using a Wald
confidence interval, the probability of detecting a disconti-
nuity with % probability and % confidence is provided by
ap/q in Equation 18. Note that ap the standard deviation
at the probability associated with discontinuity size p (see
Equation 17).
(17) σap = √
_____________________________________________
Var(ˆd) po +2 zpCov(ˆd,ˆd) po po + zpVar(ˆd)op
(18) ap/q = ap + zqˆp a (ˆd po + zpˆd) po + zqσap
Equation 19 gives 90/95 which is defined as the disconti-
nuity size with 90% POD and 95% confidence:
(19) a90/95 =ˆd po + z0.9ˆd po + z0.95σa90
When performing the conversion from a linear model to a
probability curve, the variance of the POD curve is calculated
as pod (ˆ μpod σpod) ˆ = UTV(ˆ)U where is the variance-
covariance matrix from the linear model fit, and is a matrix
of derivatives of the new pod and pod with respect to each
estimated parameter. Extending this result for a model that
estimates parameters—say, 0 , β1,…,βp−2,σε2—the matrix
has a size of × p and the matrix has a size of × p so pod
will always have a final size of × 2.
3. Methods: Alternatives to the Simple Linear
Model Setup
In step 1 of the POD process, a linear model is fit to the data,
usually the simple model shown in Equation 2. However, this
model only describes the effect that discontinuity size ( has
on the signal response ( Often, as confirmed by analysis of
variance (ANOVA), the NDE response varies with multiple
variables, not just discontinuity size, and these additional vari-
ables should be included in the linear model. For instance,
the NDE response may depend on the area of a discontinuity
rather than just its length, which may require including 2 as
a variable in the linear model. Equation 5 applies to any linear
model y however, Equation 6 becomes more complex as more
variables are added, and all steps for calculating the POD curve
will require adjustment.
3.1. Variable Additions to the Simple Linear
Model Setup
The methodology defined in MIL-HDBK-1823A [4] does not
specify how to calculate POD for models beyond the simple
linear relationship between and but additional terms
are often needed. This section considers the complication of
moving from the simple linear model, = β0 + β1x to a model
with additional terms and interactions.
For example, the categorical variable of material type may
have a statistically significant relationship to the NDE response.
In that case, the indicator variable (Equation 21) may be
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