TIME-OF-FLIGHT MEASUREMENTS FROM EDDY CURRENT
TESTS
Y. Tian1, A. Tamburrino2, S.S.Udpa1, andL.Udpa1
Department of Electrical and Computer Engineering
Michigan State University
2120 Engineering Building, East Lansing, MI 48823
2
DAEIMI, Laboratory of Computational Electromagnetics and Electromagnetic
Nondestructive evaluation
Universita degli Studi di Cassino, via G. di Biasio, 43-03043 Cassino-Italy
ABSTRACT. Data fusion techniques are based on the premise that two or more carefully
designed NDE tests are potentially capable of offering additional information concerning the test
object relative to what can be garnered from a single test. However, information from a
heterogeneous set of transducers cannot be fused unless the data is mapped onto a common
'format'. The eddy current method cannot, for example, provide time-of-flight information that
can be combined from estimates derived from ultrasonic tests. This paper presents a simple
solution to address the problem by employing the so-called Q-transform to relate diffusive fields,
such as those generated by eddy current probes, and propagating wave fields generated by
ultrasonic NDT sensors. The paper illustrates how the distance between a defect and a source can
be extracted from eddy current data generated using carefully selected excitation signals.
Numerical results and comparisons with analytical predictions are presented.
INTRODUCTION
The concept of data fusion has attracted considerable attention from the
nondestructive evaluation (NDE) community in recent years. One common strategy is to
inspect a specimen using more than one NDE method in the hope of acquiring additional
information than would be obtained from a single test. The difficulty lies in the fact that
different NDE tests are often based on different physical phenomena. As a result,
measurements from a heterogeneous sensor environment cannot usually be directly fused.
For example, time-of-flight (TOF) measurements are widely used in wave propagating
phenomena based tests, such as ultrasonic NDE tests, to obtain defect location and sizing
information. In contrast, the concept of TOF has no meaning in diffusion phenomena based
eddy current test since solutions for the diffusion equation do not allow for a well-defined
group velocity.
Instead of mapping eddy current and ultrasonic testing data onto the same format [1],
we propose to employ a Q-transform [2-5] based method to retrieve time-of-flight
information directly from eddy current measurements. Starting with the time domain
diffusion equation problem underlying the eddy current test, we construct an equivalent
CP657, Review of Quantitative Nondestructive Evaluation Vol. 22, ed. by D. O. Thompson and D. E. Chimenti
© 2003 American Institute of Physics 0-7354-0117-9/03/S20.00
593
wave equation problem in a fictitious time domain, called q domain, so that the time
domain diffusion equation problem and its fictitious time domain counterpart constitute a
Q-transform pair. Then, the (fictitious) time-of-flight information is linked (and extracted)
to some direct measurable features of the solution of the diffusion equation (in the time
domain diffusion). This is possible provided that a carefully designed excitation signal is
applied in the eddy current test. Specifically, the time-of-flight information is directly
related to the position of the peak of the eddy current response.
In a previous work [5], a method for extracting time-of-flight information from
diffusive fields was proposed with reference to a one-dimensional and to a threedimensional geometry. The three-dimensional problem involves the estimation of the
distance between a point source and a 'localized' anomaly. The scheme calls for
replacement of the anomaly with a localized source using linear Born approximation.
In this paper the validity of the approximations involved in solving the threedimensional scalar problem is numerically investigated by solving the diffusion and wave
propagation equations with axisymmetric models. In addition, other issues concerning the
numerical simulations and the design of the excitation signal are also discussed.
TIME-OF-FLIGHT FOR 3D DIFFUSIVE FIELDS
In this section the main issues concerning the extraction of the time-of-flight from
diffusive field data are summarized with reference to a scalar three-dimensional
configuration.
Q-transform
The Q-transform is an integral operator that can be defined as follows:
(1)
where x is the spatial coordinate, t is the time coordinate and q is the fictitious time
coordinate.
Let v(x,f) and w(x,#) be solutions of diffusion equation and equivalent wave
equation
V2v(x,t)-s(x)dtv(x,t)=G(x,t)
v(x,0)=0
limv(x,f)=0
in 9t3,
in 9t3
for t>0
for t>0
(2)
9
and
V2u(x,q)-s(x)dqqu(x,q)=F(x,t)
in
9t3,
3
w(x,0)=aX*,0)=0
limw(x,g)=0
x
in 9t
for q>Q
J h°°
respectively.
Under the constraint
594
for q > 0
(3)
(4)
the Q-transform maps w(x,#) onto v(x,f) by:
Extraction of Wave Front
The time-of-flight information for a wave u is a well-defined quantity that can be
directly related to the wave front of u. In particular, for a wave launched at #=0, the
waveform at the receiving point is of the type u(q)=m(y-qf} (we omit the spatial
dependence since the Q-transform does not act on the spatial coordinates), where m
vanishes for q<0 and #/is the time-of-flight.
In eddy current tests we can measure v instead of u. Our objective is to estimate the
time-of-flight qf from v by exploiting equation (5).
Specifically, if u(c[+f) is finite and different from zero, u(q) is differentiable for
q>qf, \u'(q\ is bounded by the constant M, and
M«
then qfis proportional to the square root of the position (on the time axis) of the peak of |v|
[5].
Similarly, when u contains a Dirac pulse, i.e. u(q)= a6 (q -qf )+ h(q -qf ), where h
vanishes for q<Q and \h\ is bounded by a constant MI, and
Ml « a/qf ,
(7)
then qfis also proportional to the square root of the position on the time axis of the peak of
|v [5].
Therefore, under proper conditions ((6) or (7)), the position of the peak of v gives
the time-of-flight information. It is worth noting that conditions (6) and (7) are stated in
terms of the fictitious wave u. The only way to impose (6) or (7) is to design the source
term F appearing in (3) properly as shown in the next section. Once a proper F is chosen,
the excitation signal G to be used during the eddy current test is given by (4).
SOLUTION OF AN INVERSE PROBLEM: SINGLE ANOMALY DETECTION
This section addresses the following inverse problem. We wish to estimate the
distance between a point source and an anomaly embedded in an infinite homogeneous
three-dimensional medium. We assume that the host media is a conductor of conductivity
O 0 , permeability //0 and the displacement current can be neglected. In order to show the
main ideas underlying the proposed approach, we simplify the problem by modeling it with
595
•
•Mill:
FIGURE 1. The reference geometry. A spherical homogeneous anomaly is insonified by a point source
located at the origin of the coordinate system.
a scalar diffusion equation.
Furthermore, we assume that the anomaly is confined within the ball
BR^ = jxe 9t3 | |x-x 0 | < R\ where R «|x0|, the point source is located at the origin of
the coordinates system and the measured quantity is the reaction field (evaluated at the
origin of the coordinates system) due to the anomaly. This simple canonical problem
involves the estimation of d = |x0| in Figure 1.
Design of the Waveform of the Source Term
The approach followed in designing the source terms consists of developing a
'simple' analytical model for the measured quantity in the fictitious q domain. This model
is then used to impose either constraint (6) or (7) on F and, finally, G is obtained as the Qtransform of F [5]. Notice that the design of the source terms F and G depends on the
geometry under consideration.
Following the assumption of a point source located at the origin of the coordinate
system, we have F(x,t)= d(x)f(q) and G(x,t)=d(x)g(t) where g = Q{f} . Let
^(x)= o(x)/o 0 (x)-l be the contrast function, the function s(\) appearing in (2) and (3) is
s(x)= [l + ^(x)yco , where c0 = l/^ju0(T0
is the velocity of the fictitious wave field. Notice
that % vanishes outside the region containing the anomaly, i.e. ^(x)= 0 for |x - x 0 | > R .
It is well known that the solution for the fictitious wave equation (3) is the sum of us,
the field scattered from the anomaly, and the incident field UQ , the solution of (4) for
% = 0. Similarly, the solution of the diffusion problem (2) is the sum of vs, the reaction
field due to the anomaly, and v 0 , the background field for % = 0.
Thanks to the connection (5) between the solutions of (2) and (3) we also have that
us and vs are connected via (5), i.e. vs(\,t)=Q$is(x,q)yt). Since we have assumed
vs (0,f) to be the measured data, condition (6) or (7) has to be applied to us (0,#). Using the
first-order linear Born approximation, and replacing the Born source (whose support is
contained in BR^) with an equivalent point source, us(ti,q) can be computed analytically
[5]:
596
(8)
where K =
We assume the excitation signal f(q) in the fictitious time domain is:
f(q)=(q-qijH(q-qi}
(9)
where qi > 0. It can be shown that equation (7) is automatically satisfied for n=l whereas
equation (6) is automatically satisfied for n=2.
Under these conditions, the time domain solutions are:
n=
(10)
where B = K/(47KQ\xQ\J and qff =qt +2|x0|/c0 .
The corresponding time domain excitation signals are
erfc(qjj4i)
(11)
4t
Finally, the positions of the peak values of the signal (from (10)) are given by
(12)
the peak values are
»=i
B
n =2
(13)
and the solutions Ix 0 | of the inverse problem are
(14)
x n =•
597
Numerical Simulations
This section shows the effectiveness of the proposed method and discusses the limits
of validity.
We assume that the anomaly is a ball of radius R = 0.1 mm and that the distance
between its center and the origin of the coordinate system is |xo|=10 mm. The conductivity
0 0 of the surrounding material is 3.54xl0 7 S and the relative permeability is 1 (copper).
This corresponds to a fictitious wave velocity of 0.15 m/^fs . The anomaly has a
conductivity of 2.655 xlO 7 S, which leads to % =-0.25 in BR^Q . The parameter q. is set
to be zero.
The computation of both us and vs has been carried out by exploiting the
axisymmetry of the configuration, thus making the numerical simulations much more
efficient in terms of computational effort.
The governing equations for us and vs under the premise of axisymmetry are:
a
d2us(r,z,q)
•^ 2
dr
"""
1 dus(r,z,q)
-^
r
dr
"""
d2us(r,z,q)
-^ 2
dz
1 d2us(r,z,q)
~\ 2
CQ2
dq
together with proper boundary conditions at infinity and (point source assumption):
—.
(17)
The solutions for both diffusion equation and wave equation have been obtained using the
finite element method. A zero Neumann boundary condition is imposed on the symmetry
axis while a zero Dirichlet boundary condition is imposed on the outer boundary to
simulate the infinite domain. Specifically, the numerical simulations were carried out using
a commercial code (FEMLAB®) running on a PC equipped with a 2.2 GHz Pentium IV
CPU and 512 MB physical memory. A mesh with 6517 nodes and 12778 triangles is used.
A higher mesh density is employed in the vicinity of the anomaly as shown in Figure 2.
!1M
FIGURE 2. A schematic of the geometry for the numerical simulation (left) together with the finite element
mesh (centre). Expanded view of the mesh in the region containing the anomaly (right).
Linear elements are used everywhere except in the shadowed region where quadratic elements are employed
instead.
598
A comparison between the time domain scattered field vs(to,t) and the QA comparison between the time domain scattered
scattered field
field vvss((00,,tt)) and
and the
the QQtransformed fictitious time domain response us(ft,t) was carried out for
the n = 2 case to
transformed fictitious time domain response u s (0, t ) was carried
carried out
out for
for the
the nn == 22 case
case to
to
transformed
highlight the mapping property of the Q-transform.
Fig 3 shows agreement achieved
highlight
of
the
Q-transform.
Fig
3
shows
agreement
achieved
highlight
the
mapping
property
3
shows
agreement
achieved
between the (numerical) solution of the diffusion problem (15) and the numerically
between the
of thesolution
diffusion
and
numerically
between
(numerical)
solution
problem
(15)
and the
theproblem
numerically
computed
Q-transform
of the
(numerical)
of problem
the
wave (15)
propagation
(16).
computed Q-transform of the (numerical) solution of the wave
computed
wave propagation
propagation problem
problem (16).
(16).
We notice that the peak of the time domain response (the observable quantity vs(ti,t))
is
We notice
notice that the peak of the time domain response (the observable
We
observable quantity
quantity vvss((00,,tt)))) isis
very close (2.3%) to the theoretical value of 0.009s predicted by (12). This confirms that
very close
close (2.3%) to the theoretical value of 0.009s predicted
very
predicted by
by (12).
(12). This
This confirms
confirms that
that
|x0| can be estimated using (14) once the peak position has been determined from the
x
can
be
estimated
using
(14)
once
the
peak
position
has
been
determined
from
x 00 can
has been determined from the
the
measured
waveform
v
(0,f).
s
measured waveform v ss (0, t ) .
measured
Figures 4-5 highlight the limits of
validity of equation (8)
(8) and, consequently,
consequently, ofthe
the
Figures 4-5 highlight the limits of validity of equation
equation (8) and,
and, consequently, of
of the
inversion
formula
(14).
In
particular,
Figure
4
shows
the
numerically
computed
time
inversion formula (14). In particular, Figure 4 shows the
inversion
the numerically
numerically computed
computed time
time
domain
response
vvs(to,t)
together
with
its
analytical approximation
approximation (8),
(8), for
forboth
both n = 1 and
(
)
domain
response
0
,
t
together
with
its
analytical
domain
approximation (8), for both nn ==11 and
and
ss
nn==22 cases. We notice that the relative error
affecting (14) for
for #,=0 depends
depends only on
on the
n = 2 cases. We notice that the relative error affecting (14)
(14) for qqii=0
=0 depends only
only on the
the
dimensionless
y=R/d and on %, as seen by
by using
using the
the dimensionless
dimensionless
dimensionless parameter
using the
dimensionless
dimensionless
parameter γ =ˆ R / d and on χ, as seen by
ˆ
coordinates
| and
and ttf'' ===
where T = §x \/c 22J.
coordinatesx'xx'' and
and ftt'' defined
defined by
by x'=x/|x
xx'' =
ˆˆ ttt ///TTT ,,, where
coordinates
and
defined
by
=ˆˆ xx // xx 000 and
where TT =ˆ=ˆ ( xx00 Q cc00)Q) ..
*v.
Q.Q1
0.02 ;6.Q3
0.04
0,05
Q.Q&
Oi07
0;p8
0.0!
(0,t)t ) (solid)
{u s5 ((0,
)} (*).
FIGURE3.3.
3.The
Theplot
plotof
of vvv5 s(0,
(solid) together
together with
with the
the plot
plot of
0, tt)}
FIGURE
of Q
g{w
(*). The
The predicted
predictedpeak
peakposition
position
FIGURE
The
plot
of
s (0, t ) (solid) together with the plot of Q{u s (0, t )} (*). The predicted peak position
at0.009s.
0.009s.
isisisatat
0.009s.
x.
Q.01
t(s)
FIGURE 4. Plots of v (0, t ) (*) together with the plot of the approximate response (solid). Left: case n=1.
(0, t ) (*)
FIGURE4.4.Plots
Plots of
of vv5ss(0,f)
(*) together
together with
with the
the plot
plot of the approximate response
FIGURE
response (solid).
(solid). Left:
Left: case
case n=1.
n=\.
Right: case n=2.
Right:case
casen=2.
n=2.
Right:
599
.1
0,2
0,3
44
0;5
0,6
,1
0.7
0-2
Q.3
0>4
0<5
0;6
0.7
FIGURE 5. The error affecting the predicted position (*) and amplitude (o) of the peak as function of 7 for
n=l (dashed line) and n=2 (solid line), respectively. Left: case ^ = -0.25. Right: case % = -0.99.
Figures 5 show the relative error affecting the estimate of |x0| by (14) for two
different values of contrast: % = -0.25 (Figure 5, left) and % = -0.99 (Figure 5, left).
These values of contrast have been chosen because they are representative of the range of
interest (the interval from -1 (perfectly insulating anomaly) to 0 (no anomaly)). Figures 5
show that the error in the estimate given by (14) is very small for j < 0.1. Notice that the
relative error grows faster than a linear rate for larger values of y
CONCLUSIONS
An inversion method based on time-of-flight for diffusive phenomenon has been
shown to be effective for detecting anomalies in conductive materials. The design of the
excitation waveform is critical in order to estimate time-of-flight properly. Numerical
modeling has made it possible to investigate the limits of validity of the inversion method.
Time-of-flight information from the eddy current testing data can be fused with time-offlight information data from other NDE techniques such as the ultrasonic method, to
improve the estimate.
ACKNOWLEDGEMENTS
This work has been supported by NASA, Italian Ministero delPIstruzione,
dell'Universita e della Ricerca (MIUR), Italian Space Agency and EURATOM.
REFERENCES
1. Sun, K., Udpa, S. S., Udpa, L., Xue, T. and Lord, W., in Review of Progress in QNDE,
Vol 15, eds. D. O. Thompson and D. E. Chimenti, Plenum, New York, 1996,p813.
2. Bragg, L. R. and Dettman, J. W., J. Math Analysis andApplic. 22, 261-271, 1968.
3. Lee, K. H., Liu, G., and Morrison, H. F., Geophysics, 54, ppl 180-1192, 1989.
4. Ross, S., Lusk, M. and Lord, W., IEEE Trans. Magnetics, 32, 535-546, 1996.
5. Tamburrino, A. and Udpa, S. S., "Solution of inverse problems for scalar parabolic
equations using a hyperbolic to parabolic transformation: Time Domain Analysis",
submitted for publication
6. Tamburrino, A. and Udpa, S. S., "Solution of inverse problems for parabolic
equations using the Q-Transform: Time domain analysis", Internal Report,
MSU/ECE/02-11, Michigan State University, June 2002.
600