Flaw
Reconstruction Techniques
In nondestructive evaluation of structural material defects,
the size, shape, and orientation are important flaw parameters
in structural integrity assessment. To illustrate flaw reconstruction,
a multiviewing ultrasonic transducer system is shown below. A
single probe moved sequentially to achieve different perspectives
would work equally as well. The apparatus and the signalprocessing
algorithms were specifically designed at the Center for Nondestructive
Evaluation to make use of the theoretical developments in elastic
wave scattering in the long and intermediate wavelength regime.
Depicted
schematically at the right is the multiprobe system consisting
of a sparse array of seven unfocused immersion transducers. This
system can be used to "focus" onto a target flaw in
a solid by refraction at the surface. The six perimeter transducers
are equally spaced on a 5.08 cm diameter ring, surrounding a center
transducer. Each of the six perimeter transducers may be independently
moved along its axis to allow an equalization of the propagation
time for any pitchcatch or pulseecho combinations. The system
currently uses 0.25 in diameter transducers with a nominal center
frequency of 10 MHz and a bandwidth extending from approximately
2 to 16 MHz. The axis of the aperture cone of the transducer assembly
normally remains vertical and perpendicular to the part surface.
The flaw reconstruction algorithm normally makes use of 13 or
19 backscatter waveforms acquired in a conical pattern within
the aperture. The dataacquisition and signalprocessing protocol
has four basic steps.
 Step one involves the experimental setup, the location and
focusing on a target flaw, and acquisition (in a predetermined
pattern) of pitchcatch and pulseecho backscatter waveforms.
 Step two employs a measurement model to correct the backscatter
waveforms for effects of attenuation, diffraction, interface
losses, and transducer characteristics, thus resulting in absolute
scattering amplitudes.
 Step three employs a onedimensional inverse Born approximation
to extract a tangent plane to centroid radius estimate for each
of the scattering amplitudes.
 In step four the radius estimates and their corresponding
look angles are used in a regression analysis program to determine
the six ellipsoidal parameters, three semiaxes, and three Euler
angles, defining an ellipsoid which best fits the data.
The inverse Born approximation sizes the flaw by computing the
characteristic function of the flaw (defined as unity inside the
flaw and zero outside the flaw) as a Fourier transform of the
ultrasonic scattering amplitude. The onedimensional inverse Born
algorithm treats scattering data in each interrogation direction
independently and has been shown to yield the size of ellipsoidal
flaws (both voids and inclusions) in terms of the distance from
the center of the flaw to the wavefront that is tangent to the
front surface of the flaw. Using the multiprobe ultrasonic system,
the 1D inverse Born technique is used to reconstruct voids and
inclusions that can be reasonably approximated by an equivalent
ellipsoid. So far, the investigation has been confined to convex
flaws with a center of inversion symmetry. The angular scan method
described in this paper is capable of locating the bisecting symmetry
planes of a flaw. The utility of the multiprobe system is, therefore,
expanded since twodimensional elliptic reconstruction may now
be made for the central slice. Additionally, the multiprobe system
is well suited for the 3D flaw reconstruction technique using
2D slices.
The
modelbased reconstruction method has been previously applied
to voids and incursion flaws in solids. Since the leastsquares
regression analysis leading to the "best fit" ellipsoid
is based on the tangent plane to centroid distances for the interrogation
directions confined within a finite aperture. The success of reconstruction
depends on the extent of the flaw surface "illuminated"
by the various viewing directions. The extent of coverage of the
flaw surface by the tangent plane is a function of the aperture
size, flaw shape, and the flaw orientation. For example, a prolate
spheroidal flaw with a large aspect ratio oriented along the axis
of the aperture cone will only have one tip illuminated (i.e.,
covered by the tangent planes) and afford a low reconstruction
reliability. For the same reason, orientation of the flaw also
has a strong effect on the reconstruction accuracy.
The diagram on the right shows the difference in surface coverage
of a tilted flaw and an untilted flaw subjected to the same insonification
aperture. Both the experimental and simulation studies of the
aperture effect reported before were conducted for oblate and
prolate spheroids oriented essentially symmetrically with respect
to the part surface and hence the aperture cone. From a flaw reconstruction
standpoint, an oblate spheroid with its axis of rotational symmetry
perpendicular to the part surface represents a high leverage situation.
Likewise, a prolate spheroid with its symmetry axis parallel to
the part surface also affords an easier reconstruction than a
tilted prolate spheroid. In this CNDE project, we studied effects
of flaw orientation on the reconstruction and derived a new dataacquisition
approach that will improve reliability of the new reconstruction
of arbitrarily oriented flaws.
The orientation of a flaw affects reconstruction results in the
following ways.
 For a given finite aperture, a change in flaw orientation
will change the insonified surface area and hence change the
"leverage" for reconstruction.
 The scattering signal amplitude and the signal/noise ratio
for any given interrogation direction depends on the flaw orientation.
 Interference effects, such as those due to tip diffraction
phenomena or flash points may be present at certain orientations.
Of course, interdependencies exist in these effects, but for
the sake of convenience they are discussed separately in the
following.
Aperture
To assess the effects of finite aperture size on flaws of different
orientation, computer simulations were performed for an oblate
spheroid with semiaxes of 400, 400, and 200 µm that is
tilted and untilted with respect to the part surface. For each
of the 13 scattering directions, the exact radius estimates Re
(i.e. the tangent plane to centroid distances) were first computed,
and a random error in sizing was then introduced to simulate the
experimental situation. The radius estimate used was then taken
to be
Re'
= Re ( I + n )
where n is a randomly generated number between
±0.1. Using the Re' values for the various directions,
a best fit ellipsoid is determined using a regression program.
This process is repeated 100 times for each aperture angle and
mean standard deviation of the three semiaxes is expressed as
a percentage of the expected values. The simulation was performed
for the untilted case with the 400 x 400 µm plane parallel
to the part surface and for a tilt angle of 40ƒ from the
normal of the part surface. The results are summarized in Table
I.
The mean values for the ellipsoidal semiaxes converge to expected
values, while the standard deviations converge to some asymptotic
minimum. The values in Table I show that for a small aperture,
the standard deviation as a percentage of expected value (an indication
of the reconstruction error) is much higher for the oblate spheroid
tilted at 40ƒ with respect to the horizontal than is the
0ƒ untilted case. As the aperture increases, the difference
in reconstruction error approaches zero because surface illumination
is sufficient to ensure a reliable reconstruction. Due to the
combined effect of finite aperture and a prior unknown flaw orientation,
a large aperture is desirable to increase reliability of reconstruction
results.
Note that in this simulation only the aperture angle is increased,
and the number of interrogation directions remains unchanged.
The number of look directions is kept the same because the multiviewing
system is intended for acquiring a sparse array of data based
on speed considerations.
Signal/Noise Ratio
For
a given scattering direction amplitude of the scattering amplitude
and, therefore, the signal/noise ratio depend on orientation of
the flaw. In the short wavelength limit scattering amplitude is
proportional to square root of (R1 R2) with R1 and R2 being the
principal radii of curvature of the flaw for the scattering direction
used. This dependence is found to be important in the intermediate
frequency regime as well. To illustrate this effect, the figure
at the right shows the scattered signal amplitudes from a footballshaped
prolate spheroidal void with two cusplike tips in two directions:
broadside and along the tips. The profile of the tips can increase
the ratio of the two signal amplitudes as large as 35.
To
investigate the correlation between the accuracy of flaw sizing
and signal/noise ratio of the flaw waveform at different scattering
directions, a 400 x 400 x 200 µm oblate spheroidal void
in titanium with its axis of rotational symmetry tilted at a 30ƒ
angle from normal to the part surface was reconstructed using
the multiviewing transducer system. It was found that sizing results
were generally more accurate for the scattering directions with
a higher signal/noise ratio, as expected. Furthermore, the directions
that gave the poorest signal/noise ratios were often ones closest
to being in an edgeon perspective. The figure on the right shows
the relationship between the percentage error of the radius estimate
and signal/noise ratio of the flaw waveform. Reconstruction results
of the oblate spheroid void tilted at 30ƒ are listed in Table
II.
The reconstruction results of both the semiaxes length and tilt
angle were improved by rejecting four data points with the lowest
signal/noise ratios. Since multiviewing transducer system provides
a maximum of 19 independent look angles for a given tilt angle
of the transducers, rejecting a small subset of the data points
based on signal/noise consideration still leaves a sufficient
number of data points for the ellipsoidal regression step which
requires a minimum of six data points.
Flash Point Interference
The multiview transducer system and associated signalprocessing
algorithms reconstruct a flaw based on a general ellipsoid model.
For ellipsoids with a large aspect ratio and flaw shapes that
approach those of a flat crack or a long needle, edge or tip diffractions
due to points of stationary phase (flash points) governed by,
geometric acoustics become important. When such phenomena are
present within the transducer bandwidth, the scattered signal
frequency spectrum contains strong interference maxima and minima
and renders radius estimates by the 1D inverse Born difficult
or impossible.
The figures below show a test flaw in the form of a copper wire
segment embedded in a transparent thermoplastic disk and tilted
at 45ƒ with respect to the disk surface and the frequency
spectrum of the wire inclusion at a scattering angle of 21ƒ
from the wire axis. The strong interference pattern prevented
the ID inverse Born algorithm from yielding a meaningful radius
estimate. However, when the spectrum was analyzed on assumption
of flash point interference (without having to use the angle information),
321 µm was obtained for a path length difference of the
stationary phase points in the scattering direction; this compared
reasonably well with 374 µm for twice the tangent plane
distance in this orientation.

Photomicrograph of copper wire segment titled at 45ƒ and
embedded in thermoplastic. Each minor division of scale is 10
µm, and wire segment is approximately prolate spheroid with
semiaxes Ax = 80 µm, Ay = 80 µm, and Az = 200 µm.
Spatial Dataacquisition Pattern For
Arbitrarily Oriented Flaw
From the investigation described earlier, it is clear that reliable
reconstruction of an arbitrarily oriented flaw generally requires
a large aperture. However, a large viewing aperture perpendicular
to the part surface may still contain scattering directions hampered
by weak flaw signal amplitude (poor signaltonoise ratio) and,
in certain cases, flash point interference. A predetermined dataacquisition
pattern that is relatively free from such disadvantages can improve
reconstruction reliability. In this work we explored a method
to predetermine a spatial pattern for data acquisition. This pattern
affords a high leverage for reliable reconstruction for arbitrarily
oriented flaws that can be approximated by the shape of a spheroid.
Consider
a tilted prolate spheroid as shown on the right. We may define
a vertical sagittal plane (VSP) as the plane that bisects the
flaw and contains the z axis. We further define a perpendicular
sagittal plane (PSP) as the plane bisecting the spheroid and perpendicular
to the VSP. The intersection of the VSP and PSP (direction M in
diagram) then corresponds to a direction of maximum flaw signal
amplitude. The orientation of the VSP can be located by a series
of azimuthal scans at different polar angles. A maximum in the
signal amplitude should be observed at the azimuthal angle of
the VSP. This definition of the VSP and PSP and their relationship
to backscattered flaw signal amplitude also holds true for an
oblate spheroid. Below shows the azimuthal scans at four different
polar angles for the 2.5:1 prolate spheroid (wire segment) flaw.
Once the azimuthal angle of the VSP is determined (30ƒ in
this case), a polar scan below at the azimuthal angle of the VSP
determines the tilt angle of the wire segment to be 41ƒ,
as compared to 45ƒ from optical measurement.

Flaw signal amplitude as a function of azimuthal and polar angles.
The angular scans serve two very useful functions.
First, they provide some information about the shape and orientation
of the flaw. For example, a scan in the perpendicular sagittal
plane can distinguish a prolate spheroid from an oblate spheroid
by changing the polar angle and the azimuthal angle simultaneously.
A scan in the PSP of the 2:1 oblate spheroid tilted at 30ƒ
showed a peak in flaw signal amplitude at the intersection of
the VSP and the PSP (direction M), whereas a scan in the PSP of
the tilted 2.5:1 prolate spheroid showed a constant flaw signal
amplitude.
Second, it provides a basis for predetermining a
spatial dataacquisition pattern that is equivalent to a tilted
aperture cone centered at direction M. This dataacquisition pattern
not only ensures good signaltonoise ratio, avoids possible flash
point interference due to endon or edgeon perspectives, and
provides a maximum illuminated area on the flaw surface, but also
allows one to reconstruct the flaw with two mutually orthogonal
elliptical cross sections in the VSP and PSP.
So far, the discussion of angular scans has been confined to
flaws that are approximately spheroidal in shape. For a general
ellipsoid with three unequal semiaxes and oriented arbitrarily
in space, the angular scan results will be more complicated. For
example, an azimuthal scan at different polar angles is not expected
to show a peak at the same azimuthal angle. Shape and orientation
information, in principle, can still be extracted from such data,
and further investigations are underway for the general case.
Reconstruction Results
To verify the reconstruction method using the new spatial dataacquisition
configuration experimentally, reconstructions were performed on
two test specimens. The first flaw was the 400 µm long 80
µm radius copper wire segment embedded in a thermoplastic
disk. This flaw was used to approximate a prolate spheroid with
a 2.5:1 aspect ratio. The axis of the wire segment was at a 45ƒ
angle relative to the part surface. The second flaw was a 400
x 200 µm oblate spheroidal void tilted at a 30ƒ angle
in a diffusion bonded titanium disk, as just described.
The flaw reconstruction procedure using an aperture cone perpendicular
to the part surface was first carried out for the 2.5:1 prolate
inclusion (copper wire) tilted at a 45ƒ angle. Difficulties
due to a poor signaltonoise ratio and flash point interference
associated with look directions close to the endon perspective
prevented a successful reconstruction; in fact, enough inconsistencies
occurred in the tangent plane distance estimates that the regression
step failed to converge.
Based on orientations of the sagittal planes determined in the
angular scans, the new dataacquisition pattern equivalent to
tilting the aperture axis to the direction of maximum signal strength
was used. The ellipsoidal reconstruction gave a tilt angle of
42ƒ and three semiaxes of 257, 87, and 81 µm. These
results compared very favorably with the actual tilt angle of
45ƒ and the actual semiaxes of 200, 80, and 80 µm.
The
new dataacquisition pattern also allows one to reconstruct an
arbitrarily tilted spheroidal flaw with the two mutually orthogonal
elliptical crosssectional cuts in the VSP and PSP. This was done
for the copper wire inclusion. After identifying the vertical
sagittal plane and the perpendicular sagittal plane, a series
of tangent plane distance estimates were made for scattering directions
confined in these two planes. Using these results, the two mutually
orthogonal elliptical cross sections in the VSP and PSP were reconstructed
using a similar regression program in 2D. The two reconstructed
ellipses were 266 x 83 µm and 80 x 75 µm, respectively,
and the tilt angle was found to be 51ƒ. Table III shows the
results of the 3D reconstruction using 19 look perspectives and
the 2D reconstruction of the ellipses in the VSP and PSP. Both
reconstructions compared very favorably with the expected values.
The greatest discrepancy is in the value of the semiaxis Ax;
this is to be expected because the wire segment is approximately
a prolate spheroid with two ends truncated.
The 2:1 oblate spheroidal void tilted at a 30ƒ angle in
a titanium disk was investigated, again, following the procedure
of predetermining a favorable dataacquisition pattern based on
angular scan results. Table IV shows the reconstruction results
using the new dataacquisition pattern equivalent to an aperture
cone centered on the direction of maximum backscatter signal.
As a comparison, reconstruction results using an aperture cone
normal to the part surface (described earlier) are also shown.
As can be seen, the improvement of the reconstruction by using
the new dataacquisition pattern is not as dramatic as the prolate
inclusion case. This is consistent with the fact that the oblate
spheroid has a smaller aspect ratio and a smaller tilt angle and
is therefore not nearly a "low leverage" flaw to reconstruct
using the normal (untilted) dataacquisition pattern.
The reliability problem of reconstructing arbitrarily oriented
flaws using the multiviewing transducer system and associated
modelbased algorithm has been studied. An arbitrarily oriented
flaw may afford a low leverage for reconstructing the entire flaw
based on limited surface area covered by the tangent planes in
a finite aperture and, therefore, requires a greater aperture
for a reliable reconstruction. However, the aperture size has
practical limits in a singleside access inspection situation
and a larger aperture does not necessarily alleviate such difficulties
as poor signaltonoise ratio and flash point interference associated
with certain interrogation directions. In our study of reconstructing
approximately spheroidal flaws oriented at some arbitrary angle,
it was found beneficial to predetermine a spatial dataacquisition
pattern based on angular dependence of the flaw signal amplitude.
The new dataacquisition pattern is equivalent to tilting the
interrogation aperture cone to compensate for the particular orientation
of the flaw and restore the leverage for a more reliable reconstruction.
This method worked well on two test cases.
