5th Pan American Conference for NDT · The use of Lamb waves, to perform tomography reconstruction process involves interaction of guided waves with discontinuities.

Ultrasonic Tomography Using Lamb Wave Propagation Parameters

Antonio BALVANTIN 1, Arturo BALTAZAR2

1,2 Robotics and Advanced Manufacturing Program, CINVESTAV-Unidad Saltillo, Carretera

Saltillo-Monterrey Km 13.5 Ramos Arizpe, Coah, 25900, México

Phone/Fax: +52-844-439-0305

e-mail: 1 [email protected] ;

[email protected]

Abstract

Tomography refers to the cross-sectional imagining of an object from either transmission or reflection data

collected by illuminating the object from different directions. In this work, an ultrasonic tomography system

based on Lamb waves, developed in our laboratory, is presented. We discuss the inverse problem of tomography

imaging using data collected from ultrasonic plate waves interacting with a discontinuity. Here the main focus is

on improving the quality of tomography maps using two reconstruction techniques (backprojection and MART).

Results of experiments for defect reconstruction on a thin plate are presented. It was found that MART can

provide a better reconstruction image than the backprojection technique, with minimal distortion, of a two-step

circular discontinuity embedded in an aluminum plate.

Keywords: Ultrasonic Lamb waves; tomography; defects; time-frequency representation.

1. Introduction

Defect detection and characterization are critical tasks for structural health monitoring of

engineering structures. Physical aging during routine operation inflicts deterioration and

generates hidden internal and/or superficial defects in these structures. Typically, real time

monitoring of engineering structures requires scanning large areas in a short period of time.

Lamb waves have the ability to propagate long distances in plates and pipe-like structures,

offering a potential solution to defect detection in these types of structures [1].

Several approaches have been proposed to achieve tomography reconstruction on different

types of structures. Leonard, et. al. [2], discuss the use of a square perimeter array of

transducers to achieve tomography reconstruction of aircraft structures. Malyarenko &

Hinders [3] present a methodology for tomography reconstruction with ray bending correction

for imaging of objects with moderate ultrasonic scattering. Volker et al. [4] reported the use of

ultrasonic tomography for corrosion-type discontinuities reconstruction in pipes using Lamb

wave traveltime parameter.

In this work, the application of ultrasonic tomography using Lamb waves for reconstruction

of artificial discontinuities in aluminum plates was studied. The objective in this study is to

improve Lamb wave ultrasonic tomography reconstruction of an artificial discontinuity using

5th Pan American Conference for NDT 2-6 October 2011, Cancun, Mexico

the ray tracing theory. The implementation of a parallel projection tomography scheme as a

data gathering process is used to compare amplitude reconstruction using a backprojection

algorithm with traveltime reconstruction and MART algorithm.

2. Tomography theory

Ultrasonic tomography can be studied in a similar way as x-ray tomography if wave refraction

and diffraction phenomena are ignored in a transmitter-receiver configuration of transducers

[5]. In both cases, a line integral (Figure 1) of the attenuation or traveltime values can be

estimated on the far side of the object by,

,

( , )

( ) ( , )t line

P t f x y ds

(2.1)

( , ) ,P P

P

T s x y dl (2.2)

where ( )P t is the projection amplitude as a function of positions ti, is the projection

angle, ( , )f x y is the inspected object function, ds is the distance from the projection axis

iX to the position ( , )x y in the object, PT is the travel time along an arbitrary path P that

connects a given transmitter and receiver position, ( , )s x y is the slowness distribution of the

propagation media and Pdl is an infinitesimal distance along the path .P A more detailed

description of equations (2.1) and (2.2) is presented in next sections.

In Figure 1, the generation process of a projection ( )P t from several ultrasonic signals

propagating through an inspected object, ( , )f x y , is shown. The amplitude or traveltime

values taken at multiple transmitter-receiver transducer positions and a specific angle i

generate a single projection.

In theory, if an infinite number of projections are obtained, the object function can be entirely

recovered. Tomography process involves many wave-matter interactions when input signals

travel from the transmitter to the receiver transducer and are affected by the propagation

media.

Figure 1. Implementation of the line integral, at the angle , to obtain the projection of an object ( , )f x y .

a)

b)

Figure 2. a) Lamb waves schematic, showing an h1 thick plate with a thickness reduction h2; b) Symmetric (solid

lines) and Antisymmetric (dashed lines) Lamb waves modes for a 2.03 mm thick Aluminum plate.

The use of Lamb waves, to perform tomography reconstruction process involves interaction

of guided waves with discontinuities. The interaction of guided waves with discontinuities

affects the attenuation and velocity properties of the ultrasonic signals. The problem of Lamb

wave propagation in a plate with thickness reduction (discontinuity) is schematically

described in Figure 2.

Theoretically, an infinite number of dispersion Lamb wave modes can be excited in the plate.

A discontinuity similar to that shown in Figure 2a changes the behavior of the dispersion

curves (Figure 2b). Due to dispersion phenomena, interpretation of Lamb waves and

identification of propagation parameters requires additional signal processing. Multiple

dispersion modes can be found in a typical Lamb wave signal and arrival time for specific

frequencies cannot be resolved in the time-domain representation. Recently, Short Time

Fourier Transform (STFT) has been reported for mode identification of guided waves [6].

Here, amplitude and traveltime parameters are obtained by implementing STFT analysis.

A schematic of the data generation and gathering processes performed in our experiments is

illustrated in Figure 3. Once the ultrasonic signals are captured and signal parameter

Y

X

θi

Inspected

object

Transmitter

ReceiverProjection

angle

( , ) / ( , )f x y s x y

h2h1

Transmitter Receiver

Lamb wave wavefront

Single-step circular

discontinuity

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1

2

3

4

5

6

Frequency (MHz)

Ph

ase

Vel

oci

ty (

m/m

s)

A0

S0

A0

A2 S2

A3S0

1.015 mm

2.03 mmA1

2.03 mm

1.015 mm

2.03 mm

1.015 mmA1

S1

identification process is carried out, the input data for the tomography reconstruction

algorithms can be obtained.

Figure 3. Scheme for the data gathering process for the two experimental setups.

2.1 Backprojection algorithm

In principle, to perform a tomography reconstruction, a summation of all the projections is

needed in either frequency or space domain. The sum of all projections realized in the space

domain is called the backprojection algorithm [5].

Conceptually, the complete steps of filtered backprojection algorithm are described in Figure

4.

Figure 4. Block diagram of backprojection algorithm scheme.

In this study, the backprojection algorithm to reconstruct an artificial image (shown in Figure

4) was implemented on synthetic data that simulates a two-step thickness reduction in a plate

(see Figure 5a). This is used to model a discontinuity with corrosion-like shape. Smooth

(Figure 6a) and irregular (Figure 6b) profile discontinuities could be a challenge in

tomography reconstruction process because of the scattering effects produced by Lamb wave

propagation interaction with the discontinuity.

50 cm

50

cm1

2

3

4

5

Ultra

son

ic

Sig

na

lsD

ata

Vec

tor

1

Data Vector 3

2 cm

25 cm

Circular Discontinuity

2.54 cm of diameter

Aluminum Sample Plate

Generation and

Data Gathering

process of the

ultrasonic signalsMeasure the projection

P(t) for each angle

between 0 and 180

Fourier Transform P(t)

to obtain S(w)

Weighting FunctionInverse Fourier

Transform S(w)

Summation over the

image plane

(Backprojection process)

a) b)

Figure 5. Reconstruction from a synthetic figure, white background with two circles in the center.

a) Original image; b) Reconstruction using backprojection algorithm.

a)

b)

Figure 6. Plate-like structures with discontinuities. a) Gaussian profile; b) Irregular profile.

A set of 19 projections, with 100 rays each were artificially obtained from 0° to 180° in

increments of 10° emulating the parallel projection scheme on thickness reductions (Figure

5a). The artificial discontinuity was divided in rectangular cells to generate a figure matrix;

for each projection, the value of each cell was taken into account to generate the amplitude

projection matrix.

Reconstruction results using the backprojection algorithm are shown in Figure 5b. It was

found that the backprojection algorithm is able to reconstruct the shape and thickness gradient

of the original image; however, there is a distortion (reduction in size) on the reconstructed

discontinuity of about 30%. The incorrect sizing could be critical for defect characterization

of engineering structures; if actual size of discontinuities is not correctly resolved, a good

insight on health condition of the inspected object cannot be achieved.

Straight-ray propagation of ultrasonic signals introduces errors in the image reconstruction (as

shown in Figure 7a). Thus, tomography reconstruction should account for acoustic waves

bending (curved paths) in plates with refracting objects (see Figure 7b). This phenomenon is

called ray bending [7]. In order to improve tomography reconstruction and to overcome the

ray bending problem, a ray tracing algorithm is proposed. The basic idea of ray theory is to

calculate the ray path, with minimum traveltime, of ultrasonic signals from the transmitter to

the receiver transducer. The implementation of ray tracing theory uses a reconstruction

algorithm with traveltime input parameter taken from ultrasonic signals. The fundamental

theory of traveltime tomography reconstruction algorithm is described next.

20

40

60

80

100

20 40 60 80 100

20

40

60

80

100

20 40 60 80 100

h2h1

Two-steps circular

discontinuity

h3

h2h1

Transmitter Receiver

Gaussian profile

discontinuity

h2h1

Transmitter Receiver

Irregular profile

discontinuity

a)

b) Figure 7. a) Straight ray tomography; b) Ray bending problem.

2.2 MART algorithm

Several ray tracing algorithms have been developed to overcome the ray-bending problem in

tomography reconstruction. Anderson and Kak [8] presented a review of digital ray-tracing,

using the Fermat principle and eikonal equation, for applications in ultrasonic computerized

tomography.

In this study, a Multiplicative Algebraic Reconstruction Technique (MART) [7] for traveltime

tomography reconstruction is used. To achieve traveltime tomography reconstruction of

ultrasonic signals, it is necessary to reconstruct the slowness distribution ( , )s x y in the

propagation media at certain location. The slowness distribution is the inverse of Lamb wave

phase velocity ( , )pc x y in propagation media at point ( , )x y . Data for slowness distribution

calculation is obtained by traveltime parameter determination for each transmitter-receiver

positions.

The time of flight (TOF) of an ultrasonic signal from a transmitter transducer position to a

receiver transducer position can be obtained by applying the following line integral,

( , ) .P P

P

T s x y dl (2.3)

Theoretically, the slowness distribution ( , )s x y can be calculated by implementing the

following equation,

1( , ) ( , ) ,

ˆi

m i j m i j m

i

ts x y s x y

t (2.4)

where

2

1

1

ˆ, ( , ) , 1 ' .1

Iij

m im m i j ij ij ijIi

ij

i

lt s x y l l y x

lI

(2.5)

y

x

θ

Inspected object

Transmitter

Receiver Projection angle

y

x

θ

Inspected object

Transmitter

Receiver Projection angle

From equations (2.4) and (2.5) the ( , )i jx y parameter correspond to the cell index in the

discrete representation of the slowness distribution, ijl is the traversed distance over the

( , )i jx y cell, it is the measured TOF for the receiver-transducer position, imt is the minimum

TOF obtained for the generated slowness distribution with minimum TOF ( , )m i js x y and x

is the size of the sample grid to be used to reconstruct the image. The sub-index m in

equations (2.4) and (2.5) refers to the parameters obtained when the minimum TOF for an

ultrasonic ray path is found.

If ray paths are unknowns, straight ray tomography cannot be applied; therefore an iterative

process to compute an approximate slowness distribution is required [7]. As a first step to

tomography reconstruction, we assume 1m in eq. (2.5) (straight ray estimation), and obtain

an initial slowness distribution estimation. Then, ray tracing algorithm can be applied to trace

the ray paths for all transmitter-receiver positions. Once the ray paths are obtained, bend ray

MART reconstruction can be achieved solving equations (2.4) and (2.5). Finally, the ray

tracing algorithm is applied iteratively until a desirable image quality is achieved. The

minimization criteria to determine the appropriate quality of reconstruction is obtained

following equation.

,e a (2.6)

where a is an error criteria value and

1( , ) ( , ),m i j m i je s x y s x y (2.7)

the parameter e is the error value between the current slowness distribution calculation

1( , )m i js x y and the previous one ( , ).m i js x y

a) b)

Figure 8. Reconstruction from a synthetic figure, white background with two circles in the center.

a) Original image; b) Reconstruction using backprojection algorithm.

The same synthetic image (Figures 5a) used for numerical validation of the amplitude

reconstruction process was used to validate the traveltime reconstruction. Again, a set of 19

projections were artificially obtained from 0° to 180° with increments of 10° using the

20

40

60

80

100

20 40 60 80 100

20

40

60

80

100

20 40 60 80 100

parallel projection scheme. For each projection the sum of each row was obtained to simulate

the TOF vector.

The reconstruction of the synthetic figure with MART algorithm is shown in Figure 8. The

results indicate that MART algorithm can reconstruct the shape and the two step features of

the original images with an error in size prediction of 5% with respect to the original one.

3. Description of experiments and results

The experimental setup to study the sensibility of the proposed tomography reconstruction

algorithm is described in Figure 9. An aluminum plate (50 x 50 cm) with a thickness of 2.03

mm was used. A circular discontinuity (25mm in diameter) was artificially machined in the

sample plate (Figure 9). For the experiments, two broadband longitudinal normal contact

transducers with a central frequency of 1 MHz were used to induce the A1 dispersion mode in

the plate. The two transducers were set in a pitch-catch configuration separated by a distance

of 25cm between them. A set of 5 rays (to form a projection) of ultrasonic signals was

obtained by a simultaneous and parallel displacement of the transducers in steps of 2 cm

(Figure 3). Then, four projections were captured following a 45 degrees clockwise rotation of

transducers around the discontinuity.

Figure 9. Experimental system setup of a plate with an artificially machined circular discontinuity.

Signals were further post-processed by applying short-time Fourier analysis [6]. The Fourier

transform spectrum can determine which frequencies are present in the detected signal but it

does not provide information of time localization of the signal. The STFT evaluates the time-

domain signal into a series of small sections of the signal. Then, each of these sections is

windowed and the Fourier transform is applied individually for each one. Both parameters

amplitude and time can be obtained from the captured signals and can be used as an input for

the tomography reconstruction algorithms. In Figure 10, a typical time-domain representation

of captured signals for ray #3 (ray crossing the center of the loaded area, see Figure 3) is

shown.

Reconstruction of the artificial discontinuity machined on the aluminum plate was obtained

with limited data (4 projections with 5 rays each). In Figure 11a, results of backprojection are

shown. It is possible to observe that, even for a small number of rays and projections, the

discontinuity can be reconstructed. A clear improvement in resolution (Figure 11b) using

MART was found where a better agreement in shape and size of the circular discontinuity

reconstruction (white area in the center of the figure) was found. Also there is a reduction of

artifacts, noise around the discontinuity reconstruction when compared with backprojection

results.

PC: Data gathering and

post-processing process

P/R: Generation and

Capture of the signals

50 cm

Transmitter

Circular Discontinuity 2.54 cm

2.0

3 m

m

Receiver

1.0

15

mm

Figure 10. Time-domain signals obtained using longitudinal contact transducers.

a)

b)

Figure 11. Reconstruction from ultrasonic Lamb wave captured signals, using parallel projection

tomography scheme. a) Amplitude and; b) TOF measurements.

4. Conclusions

The effect of bending of ultrasonic rays in tomography reconstruction due to changes in phase

velocity and changes in thickness was accounted for by using the MART algorithm. MART

algorithm using time of signal data was proved to be more accurate than backprojection using

only amplitude. The improvement was verified on synthetic data and on actual data taken

from a plate with an artificial discontinuity. Finally, the proposed MART algorithm could be

potentially applied for the reconstruction of defects such as corrosion which reduces the

thickness of plate-like structures.

Acknowledgements

The authors want to thank CONACYT for its financial help through project CB-134564.

References

[1] J L Rose, ‘Ultrasonic waves in solid media’, Cambridge University Press, 1999.

[2] K R Leonard, E V Malyarenko and M K Hinders, ‘Ultrasonic Lamb wave tomography’

Inverse Problems, Vol. 18, 2002, pp. 1795-1808.

0 1 2 3 4 5

x 10-4

-0.4

-0.2

0

0.2

0.4

Time, sec

Am

pli

tud

e, V

Part of the Signal Correspondingto A1 mode

Ultrasonic Computed Tomogram

1 2 3 4 5 6 7 8 9

1

2

3

4

5

6

7

8

9

1 2 3 4 5 6 7 8 9 10 11 121

2

3

4

5

6

7

8

9

10

11

12

[3] E V Malyarenko and M K Hinders, ‘Ultrasonic Lamb wave diffraction tomography’

Ultrasonics, Vol. 39, 2001, pp. 269-281.

[4] A Volker, A Mast and J Bloom, ‘Experimental results of guided wave travel time

tomography’, AIP Conf. Proc., Vol., 1211, 2010.

[5] A C Kak and M Slaney, ‘Principles of computerized tomography imaging’, IEEE, New

York.

[6] C Treesatayapun, A Balvantín, A Baltazar and J Y Kim, ‘Thickness determination of a

plate with varying thickness using an artificial neural network for time-frequency

representation of lamb waves’, Quantitative NonDestructive Evaluation Conference,

Chicago, Il., USA, 2008.

[7] B Wang, J Takatsubo, and N Toyama, ‘An Improved Ray Tracing Algorithm for

Ultrasonic CT in Nondestructive Inspections’, ICSP Proceedings, 2006.

[8] A H Anderson and A C Kak, ‘Digital ray tracing in two-dimensional refractive fields’,

J. Acoust. Soc. Am, Vol 72, No. 5, 1982.

[9] S Khare, M Razdan, N Jain, P Munshi, B V Soma Sekhar and K Balasubramaniam,

‘Lamb Wave Tomography Reconstruction Using Various MART Algorithms’ Proc.

National Seminar on Non-Destructive Evaluation, Hyderabad, 2006.

[10] A Balvantín and A Baltazar, ‘A study of guided wave propagation on a plate between

two semi-spaces with imperfect boundary conditions’, Submitted for publication to

International Journal of Mechanical Science, 2010.

[11] A Balvantín, A Baltazar and J-Y Kim, ‘Ultrasonic lamb wave tomography of non-

uniform interfacial stiffness between contacting solid bodies’, AIP Conf. Proc., Vol.

1211, 2010, pp. 1463-1470.