Sabbagh 1988 Auld and Moulder 1999 Liu et al. 2008 Yusa
2009 Sabbagh et al. 2013). These approaches require accurate
forward models and a process that iteratively compares the
simulated and measurement data to adjust the model parame-
ters until agreement via a desired error metric is reached.
Several key steps were necessary to achieve the vision of
model-based inversion to characterize cracks in metallic mul-
tilayer fastener sites using BHEC techniques. The critical first
step was the development of accurate forward models using
VIC-3D® (Sabbagh et al. 2013) that precisely represented the
split-D differential coil eddy current measurement response
to notches and cracks (Aldrin et al. 2014). We note that other
researchers have developed models pertinent to BHEC inspec-
tions (Mandache et al. 2011 Bao et al. 2025), though those were
applied primarily to probability of detection (POD) evaluations.
A robust inversion scheme was then developed, leveraging
fast surrogate models incorporating simulated results (Sabbagh
et al. 2013). A comprehensive approach was created to perform
model-based inversion of cracks and electrical discharge
machining (EDM) notches using eddy current techniques with
an automated BHEC system for various discontinuity scenar-
ios (Aldrin et al. 2016). While prior work demonstrated prom-
ising capability for sizing EDM notches with varying aspect
ratios, some model discrepancy was discovered that resulted
in greater inversion error for smaller corner discontinuities
(Aldrin et al. 2019). Sensitivity to probe liftoff and the calibra-
tion process was also observed during experimental testing,
indicating a need to better compensate for variations in probe
state and adjacent material conditions (Aldrin et al. 2019).
We note that some recent work by other researchers (e.g.,
Hughes and Drinkwater 2021) has investigated model inversion
schemes for sizing the depth of EDM notches. However, only
limited progress has been made to address the sizing of cracks
in complex geometries and under real-world test operations
and conditions (Oneida et al. 2017).
In this paper, progress is presented in addressing out-
standing performance gaps and subsequently validating the
capability to characterize fatigue cracks in metallic multi-
layer fastener sites using BHEC techniques enabled by mod-
el-based inversion. Improvements are presented that enhance
the reliability of model calibration, liftoff compensation, and
inversion steps over a wider range of conditions. While earlier
demonstrations focused solely on the inversion of cracks in
high-electrical-conductivity aluminum, this study extends the
capability to address cracks in titanium (lower conductivity)
and stainless steel (moderate conductivity with permeability
greater than 1). In addition, an approach has been developed
that performs reliably over a wide range of fastener hole diam-
eters and the multiple frequencies used in common aerospace
BHEC inspections.
1.2. Assessing Uncertainty in Sizing Capability
To test the reliability of an NDE sizing capability, validation
procedures are needed to ensure the accuracy of the material
characterization techniques. Comprehensive probability of
detection (POD) evaluation procedures have been developed
to validate the reliability of NDE techniques and are used by
the US Department of the Air Force (DAF) in support of the
Aircraft Structural Integrity Program (ASIP) (US Department
of Defense 2009 ASTM 2021). While standard practices exist
for evaluating measurement system capabilities (ASTM 2014),
there are limits to their direct application for assessing NDE
characterization performance. Prior work has introduced pro-
cedures and performed demonstrations that evaluate NDE
sizing capabilities, specifically for discontinuities in welds (Førli
et al. 1998 Lozev et al. 2005). One frequently cited metric is the
calculation of the 95% safety limit against undersizing (LUS)
bound for quantifying sizing performance. However, care must
be taken with this metric, as key assumptions—such as linear-
ity in the parameters and constant variance with varying crack
size—must be verified.
A recent paper by Aldrin and Forsyth (2017) outlined
specific challenges in sizing studies and presented best prac-
tices for evaluating NDE characterization capabilities. These
approaches are applied in this work.
Another critical point for NDE sizing performance studies
is that the discontinuity state must be very well understood.
Uncertainty in the dimensions of any property or dimension
under test must be far less than the expected performance of
the NDE characterization technique. In an ideal experiment, all
epistemic uncertainty in the NDE response would be explained
by fixed variables (e.g., crack size), leaving only the natural
uncertainty due to aleatory noise. If the aleatory noise is large
compared to the NDE response, the “signal” may be lost in the
noise. However, if the crack size is inaccurately known, then
there will be more epistemic uncertainty in the relationship
between the response and crack size, as this uncertainty stems
from a lack of knowledge (e.g., imprecise measurements of
depth). This can lead to unexpected problems in modeling the
NDE response–crack size relationship, such as spurious trends
(e.g., small cracks being undersized and large cracks oversized)
or added variability (e.g., if sizing error is not dependent on
crack size).
It is important to note that many POD crack sets are inad-
equate for use in sizing demonstrations. A study by Shell et al.
(2015) investigated eddy current model-based inversion predic-
tions for a POD set of cracks in IN-100 alloy. While the length
estimates were found to be fairly accurate, the depth estimates
showed significant scatter compared to the assumed depths of
the crack set. Even when the mean relationship between crack
length and depth was perfectly understood, significant depth
variation was found in real cracks of similar lengths within
the set. Although variability and uncertainty in discontinuity
dimensions do impact POD evaluations to some degree, the
level of observed variability in this set was unacceptable for use
in crack depth sizing studies. This paper highlights the impor-
tance for the design, manufacture, and verification of fatigue
crack specimens where depth uncertainty is well-controlled,
which is necessary for a comprehensive crack sizing perfor-
mance evaluation.
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 43

2. Approach
This section presents details on the BHEC data acquisition
and key features for classification, as well as the design of the
forward model and surrogate model build for the program.
It also gives details on the entire data evaluation process for
model-based inversion, with a focus on the liftoff compensa-
tion process.
2.1. BHEC Data Acquisition and Features
An example of a BHEC indexed hole scan using an eddy
current inspection system with rotation and depth position
indexing is shown in Figure 1, for a hole with a 1.98 × 2.34 mm
top corner crack in aluminum. The scanner indexes the eddy
current probe in both the circumferential (scan) direction
(Figure 1a) and the vertical (index) direction (Figure 1b), pro-
viding full coverage of the hole surface. The vertical channel
differential probe C-scan response (shown in Figure 1c) is used
to make calls for crack indications.
One notable feature of the eddy current system used is its
ability to acquire absolute coil eddy current response data,
providing indications of where material layers start and end, as
well as the material state (Figure 1d).
For inverting the eddy current data, the C-scan data is
reduced to a pair of characteristic vectors for each channel of
data in the rotational scan (Figure 1e) and hole depth index
(Figure 1f) directions. First, the maximum response in the hole
bore (z) direction is evaluated, providing a correlated response
with the crack length and crack profile (shape). Second, a char-
acteristic eddy current response in the hole circumferential
direction is extracted through the crack/notch peak response
in z. Both vertical (Vy) and horizontal (Vx) components of the
eddy current response are recorded, resulting in four data
vectors being used for inversion.
2.2. BHEC Forward Model and Surrogate Model Build
A model of the BHEC split-D differential probe was imple-
mented in VIC-3D® (Sabbagh et al. 2013). The outer coil
radius, D-coil outer radius, D-core outer radius, and D-core
spacing were 0.86 mm, 0.71 mm, 0.64 mm, and 0.16 mm,
respectively, and the ferrite D-cores were assigned a relative
permeability of 2000. A complete split-D differential probe
model with ferrite cores was created using a 32 × 32 × 4 grid
mesh. Models were required for the three classes of discon-
tinuity types/locations typically found in BHEC inspections:
mid-bore cracks, through-thickness cracks, and corner cracks,
ME
|
CRACKSIZING
b
x (y out of plane)
z
a
b
a
b
a
y
Figure 2. BHEC discontinuity-type categories for the model build: (a) mid-bore, (b) through-thickness, and (c) corner crack/notch (d) 3D diagram of
the through-thickness notch model with edges to be meshed.
0.0
5
0
–5
1.3
2.5
3.8
5.1
6.4
7.6 0 45 90 135 180
Angular position (degree)
225 270 315 360
7
6
5
4
3
2
1
0.0
1.3
2.5
3.8
5.1
6.4
7.6 0 45 90 135 180
Angular position (degree)
225 270 315 360
0
0
5
–5
–10 45 90 135 180
Angular position (degree)
225 270 315 360
0
0.0
1.3
2.5
3.8
5.1
6.4
7.6 2 4 6
Response (V)
8 10
Z Vertical
Horizontal
Vertical
Horizontal
θ
Figure 1. Diagrams of bolt-hole eddy current (BHEC) scans of a near-surface corner crack: (a) circumferential (​ ​ (b) depth (z). C-scan responses
from the eddy current inspection system for a 1.98 × 2.34 mm top corner crack in an aluminum layer: (c) differential (vertical) coil (d) absolute coil
(amplitude). Differential response feature vectors at (e) peak circumferential and (f) depth curves for vertical and horizontal components.
44
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
Depth
in
hole
(mm)
Depth
in
hole
(mm)
Response
(V)
Depth
in
hole
(mm)