(54)
∂μ??? ̂
∂β0
=−
1
β1
∂σ???̂
β0
=0
∂μ??? ̂
∂β1
=−
ydec − β0
β1 2
=−
μ???̂
β1
∂σ??? ̂
∂β1
=
−σε
β1 2
=−
σ???̂
β1
∂μ???̂
∂σε2
=0
∂σ??? ̂
∂σε2
=
1
2σεβ1
The SAS linear model (PROC REG) returns the variance-covariance matrix in Equation 55, so the matching derivatives
matrix is in Equation 56, and the transition matrix is in Equation 57.
(55) ? =
[
Var(β0) Cov(β0, β1) Cov(β1, σε)2
Cov(β0, β1) Var(β1) Cov(β1, σε)2
Cov(β0, σε) 2 Cov(β1, σε) 2 Var(σε) 2 ]
(56) ? =
[
∂(μ???) ̂
∂(β0)
∂(σ???)̂
∂(β0)
∂(μ???) ̂
∂(β1)
∂(σ???)̂
∂(β1)
∂(μ???) ̂
∂(σε2)
∂(σ???)̂
∂(σε2) ]
=
[
−
1
β1
0
−
μ??? ̂
β1
−
σε
β1 2
0
1
2σεβ1]
=−
1
β1
[
1 0
μ??? ̂ σ???̂
0 −
1
2σε]
Thus, the conversion matrix is:
(57) ???𝒅 =[
Var(μ???) ̂ Cov(μ???, ̂ σ???)̂
Cov(μ???, ̂ σ???) ̂ Var(σ???)̂
]
where
Var(μ???) ̂ =
1
β1 2 (Var[β0] +2μ???Cov[β0, ̂ β1] +μ???2Var[β1])̂
Var(σ???) ̂ =
1
β2
1
(σ???Cov[β0, ̂ β1] +μ???σ???Var[β1] ̂̂ −
1
2σε Cov[β0, σε2] −
μ??? ̂
2σε Cov[β1, σε2])
Cov(μ???, ̂ σ???) ̂ =
1
β1 2 (σ???2Var[β1] ̂ +
1
4σε2 Var[σε2] −
1
β1 Cov[β1, σε2])

ABSTR ACT
This study aimed to probabilistically evaluate the
size of a fatigue crack on a type 304 austenitic
stainless steel flat plate using eddy current signals.
Three fatigue cracks, with depths ranging from
2 mm to 5 mm, were introduced into the plates
through a four-point bending test. After the starter
notches for the test were removed, eddy current
testing was conducted using a differential-type plus
point probe at a frequency of 200 kHz to collect
signals caused by the cracks. The fatigue crack was
modeled as a rectangular continuous domain with
a constant width and uniform electromagnetic
properties. Since mechanical damage transforms the
austenitic phase into the martensitic phase, both
the conductivity and permeability of the domain
were explicitly considered. The depth and length of
the cracks were evaluated using a Bayesian-based
inverse algorithm, assuming the electromagnetic
properties of the crack were either known (equal to
air) or unknown. When the crack was modeled as
air, the evaluated crack sizes deviated considerably
from the actual sizes. In contrast, assuming the
electromagnetic properties to be unknown provided
better evaluations with quantified uncertainty.
KEYWORDS: electromagnetic testing, cracking, finite element
simulation, uncertainty, profile likelihood
1. Introduction
Eddy current testing (ECT) is a conventional nondestruc-
tive testing method that utilizes eddy currents induced by
time-dependent electromagnetic fields. It has several attractive
characteristics, such as being contactless and highly sensi-
tive to surface discontinuities thus, it is especially effective in
detecting discontinuities on the surface of metallic compo-
nents. However, sizing a detected discontinuity from measured
eddy current signals is challenging because the signals do not
provide direct information on the discontinuity’s profile—
especially its depth—unlike ultrasonic testing. Therefore,
many studies have been performed to develop computational
inversion algorithms to evaluate discontinuity profiles from
measured eddy current signals (Bowler et al. 1994 Auld and
Moulder 1999 Chen et al. 2004 Chen et al. 2009).
One of the major challenges in evaluating the profile of
a discontinuity from eddy current signals using a computa-
tional inversion algorithm is the ill-posedness of the problem.
Usually, the algorithm postulates that discontinuities with
similar profiles produce similar signals. However, in reality,
two discontinuities whose profiles are largely different can
sometimes produce very similar signals (Yusa et al. 2006).
Consequently, the presence of even small noise, which is inevi-
table in actual inspections, could lead to a large error in profile
evaluations (Yusa et al. 2007a Yusa and Hashizume 2017).
This highlights the necessity of taking into consideration the
reliability of discontinuity profile evaluations, specifically the
potential errors in the estimated profiles.
To address this issue, recent studies have proposed more
sophisticated algorithms to evaluate discontinuity profiles
probabilistically—that is, by estimating discontinuity parame-
ters not as single values (i.e., point estimation) but as a prob-
ability density function (Cai et al. 2018 Tomizawa and Yusa
2024). When the probability density function provided by the
algorithm is locally distributed, the results would be reliable.
Conversely, a function with a large distribution indicates less
reliable results. A previous study by the authors demonstrated
that the algorithm could reasonably quantify the reliability
uncertainty in evaluating the length and depth of a rectangular
slit machined into an austenitic stainless steel plate.
This article reports further development of the algorithm
in dealing with a more practical problem: estimating the
NDTTECHPAPER
|
ME
PROBABILISTIC SIZING OF A FATIGUE CRACK
ON TYPE 304 AUSTENITIC STAINLESS STEEL
FROM EDDY CURRENT SIGNALS
TAKUMA TOMIZAWA† AND NORITAKA YUSA†*
† Department of Quantum Science and Energy Engineering, Graduate School of
Engineering, Tohoku University, Sendai, Japan
*Corresponding author: noritaka.yusa.d5@tohoku.ac.jp
Materials Evaluation 83 (8): 73–80
https://doi.org/10.32548/2025.me-04519
©2025 American Society for Nondestructive Testing
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 73