Guided wave phase velocity measurement using multi-emitter and multi-receiver arrays in the axial transmission configuration.

TLDR: This technique enables to recover accurately guided wave phase velocity dispersion curves and was designed to overcome limitations of spatio-temporal Fourier transform for receiver array of limited spatial extent as in the case of clinical assessment of cortical bone in axial transmission.

Abstract: This paper is devoted to a method of extraction of guided waves phase velocities from experimental signals. Measurements are performed using an axial transmission device consisting of a linear arrangement of emitters and receivers placed on the surface of the inspected specimen. The technique takes benefit of using both multiple emitters and receivers and is validated on a reference wave guide. The guided mode phase velocities are obtained using a projection in the singular vectors basis. The singular vectors are determined by the singular values decomposition (SVD) of the response matrix between the two arrays in the frequency domain. This technique enables to recover accurately guided wave phase velocity dispersion curves. The SVD based approach was designed to overcome limitations of spatio-temporal Fourier transform for receiver array of limited spatial extent as in the case of clinical assessment of cortical bone in axial transmission.

Figures

Fig. 3. (color online) Normalized singular values in dB scale versus frequency-thickness fe (MHz.mm) for a 2 mm thick copper plate: the central frequency of the probe is equal to 1 MHz (a) and 2 MHz (b). The number of experimental singular values (3 or 8) corresponds to the number of emitters. The signal-to-noise threshold t1 is shown with a curved line. The singular values above the threshold define the signal subspace.

Fig. 1. Geometry of the problem: the emission and reception arrays are in contact with the metallic plate.

Fig. 6. (color online) Evaluation of the guided modes phase velocities at a fixed frequency: (a) fe = 1.6 MHz.mm (1 MHz probe) and (b) fe = 3.8 MHz.mm (2 MHz probe). The moduli of

Fig. 2. The evaluation domain is shown outside the dashed domain. The resolution condition is given by Eq. (11): two velocities are resolved when their difference Δc/c is larger than the ratio λ/LR shown as linear function. The boundaries are shown on the experimental results (Fig. 4).

Fig. 5. Fitting error (%) given by Eq. (12) as a function of cL and cT for a fixed thickness e, for a 2 mm thick copper plate, evaluated with probes of central frequency f0 equal to 1 MHz (a) and 2 MHz (b), respectively.

Fig. 4. (color online) Phase velocities for 2 mm thick copper plate, evaluated with probes of central frequency f0 equal to 1 MHz (a) and 2 MHz (b). Theoretical Lamb values are shown in continuous (symmetric modes Sn) and dashed lines (anti-symmetric modes An). Experimental values, obtained with the maxima of the Norm function [Eq. (9)], are shown in dot. The phase velocities obtained with the spatio-temporal Fourier transform are marked with circles. The evaluation domain limit is shown with respect to Fig. 2. The ratios λ/LR 1/7, 1/4, 1/2, 1 and 2 are shown in thin dashed lines. The vertical lines show the domains discussed in Fig. 6.

Full text

HAL Id: hal-01394310
https://hal.sorbonne-universite.fr/hal-01394310
Submitted on 9 Nov 2016
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Guided wave phase velocity measurement using
multi-emitter and multi-receiver arrays in the axial
transmission conguration
Jean-Gabriel Minonzio, Marilyne Talmant, Pascal Laugier
To cite this version:
Jean-Gabriel Minonzio, Marilyne Talmant, Pascal Laugier. Guided wave phase velocity measurement
using multi-emitter and multi-receiver arrays in the axial transmission conguration. Journal of
the Acoustical Society of America, Acoustical Society of America, 2010, 127 (5), pp.2913 - 2913.
�10.1121/1.3377085�. �hal-01394310�

Minonzio et al. JASA
1
Guided wave phase velocity measurement using multi-emitter and
multi-receiver arrays in the axial transmission configuration
Jean-Gabriel Minonzio
a)
, Marilyne Talmant, Pascal Laugier
UPMC Univ Paris 06, UMR 7623, LIP,
15 rue de l’école de médecine F-75005, Paris, France
a)
electronic mail: jean-gabriel.minonzio@upmc.fr
running title: guided wave velocity measurement using arrays in contact
30 June 2009
5 March 2010

Minonzio et al. JASA
2
Abstract
This paper is devoted to a method of extraction of guided waves phase velocities from
experimental signals. Measurements are performed using an axial transmission device
consisting of a linear arrangement of emitters and receivers placed on the surface of the
inspected specimen. The technique takes benefit of using both multiple emitters and receivers
and is validated on a reference wave guide. The guided mode phase velocities are obtained
using a projection in the singular vectors basis. The singular vectors are determined by the
singular values decomposition (SVD) of the response matrix between the two arrays in the
frequency domain. This technique enables to recover accurately guided wave phase velocity
dispersion curves. The SVD based approach was designed to overcome limitations of spatio-
temporal Fourier transform for receiver array of limited spatial extent as in the case of clinical
assessment of cortical bone in axial transmission.
PACS numbers
4320Ye Measurement methods and instrumentation,
4380Vj Acoustical medical instrumentation and measurement techniques
4320Mv Waveguides, wave propagation in tubes and ducts
4360Fg Acoustic array systems and processing, beam-forming

Minonzio et al. JASA
3
I. INTRODUCTION
Different quantitative ultrasound (QUS) techniques are currently developed for clinical
assessment of human bone status.
1-3
The goal is to determine ultrasound-based indicators of
bone strength, suitable for the discrimination of osteoporotic patients from healthy subjects.
4
Alternatively, the possibility of bone properties determination from ultrasound measurements,
for instance cortical thickness or elasticity, has been explored.
5-7
One of the most promising
recent development in this field is the so called “axial transmission” technique: a set of
emitter(s) and receiver(s) are linearly arranged on the same side of a skeletal site, for instance
the forearm
or the leg.
8-10
The signals obtained at the receiver(s) are the combination of all
waves propagating axially along the long axis of the bone. A few studies indicate that cortical
bones support guided waves propagation, despite absorption and heterogeneity in geometry
and elasticity.
11
In one development of the axial transmission technique, attention was focused in determining
in the time domain the velocity of the first arriving signal, denoted FAS.
12,13
The FAS can be
interpreted as a guided S
0
wave in the low frequency regime (i.e., low cortical thickness-to-
wavelength ratio) and as a lateral compression wave in the high frequency regime (i.e., high
cortical thickness-to-wavelength ratio).
14
Alternatively, other studies have focused attention
on the most energetic contribution arriving after the FAS. Signal processing techniques were
proposed in vitro to isolate this signal component and to measure its phase velocity: group
velocity filtering technique prior to spatio-temporal Fourier transform
15,16
or the singular
value decomposition in the space-time domain.
17,18
In in vitro measurements, this contribution
is identified with the A
0
plate guided mode, or its counterpart for the tube model, the F(1,1)
flexural mode.
Moreover, higher order guided modes have been experimentally identified on in vitro
animal bones. Different techniques were used to extract guided mode phase or group

Minonzio et al. JASA
4
velocities. The L(0, n) longitudinal tube wave phase velocities (1 ≤ n ≤ 3) were measured on a
bovine tibia.
19
Authors used angled beam transducers and analyzed the received signals with
the phase spectrum method.
20
The method required a minimal distance between the emitter
and receiver of about 70 mm and a scanning length of about 80 mm. On a sheep tibia, guided
waves were also experimentally observed.
21
Authors used two fixed transducers and obtained
group velocities using time-frequency analysis.
22
Results were interpreted as Rayleigh-Lamb
guided modes, A
n
and S
n
(0 ≤ n ≤ 2). However an accurate model is required to interpret the
results.
The clinical relevance of ultrasound velocities to reflect specific aspects of bone strength
such as cortical thickness, stiffness, porosity is partly established. It was found in several
clinical studies that the FAS velocity discriminates healthy subjects from osteoporotic
patients.
23-25
In addition, a FAS velocity database, measured in vivo, was used in a technique
of elasticity identification.
7
From A
0
phase velocity data, inversion schemes were proposed to
estimate cortical thickness.
5
However, the A
0
guided wave presents a strong coupling with the
surrounding soft tissues which reduces its sensitivity to bone properties in clinical
measurement.
26
Currently, there is no clinical measurement of higher order guided waves. There is a
need to implement a robust method to extract guided mode phase velocities adapted to clinical
requirements. Particularly, as the probe dimensions are limited by the accessibility to the
skeletal site or bone and soft tissue, classing signal processing methods
16,18,20,22
can not be
directly applied. Thus, the aim of this paper is to introduce a method of extraction of guided
mode phase velocities from in vivo signals. Our approach is based on the singular value
decomposition (SVD) applied to the configuration of multi-emitter and multi-receiver arrays
in axial transmission geometry. SVD is a widely used filtering tool. The interpretation of the

Frequently asked questions

Q1. What is the velocities of the experimental guided modes?

The phase velocities are obtained using the received signal subspace defined by the singular vectors of the experimental response matrix. 

Q2. What are the contributions mentioned in the paper "Guided wave phase velocity measurement using multi-emitter and multi-receiver arrays in the axial transmission configuration" ?

Minonzio et al. this paper proposed a guided wave phase velocity measurement using multi-emitter and multi-receiver arrays in the axial transmission configuration using a probe consisting of two separated emission and reception arrays in contact. 

Q3. What are the values of the Lamb phase velocities?

The values of the Lamb phase velocities depend on three physical parameters: thethickness e and the transverse and longitudinal velocities, denoted cT and cL.32 

Q4. What is the number of experimental singular values n?

The number ofexperimental singular values σn is equal to minimum size of the arrays, i.e. the minimum between NE and NR, equal in the following to NE. 

Q5. What is the resolution condition for the S2 backward wave?

Following the resolution condition [Eq. (11)], two velocities are resolved when the ratio λ/LR is less thanthe velocity difference Δc/c. 

Q6. What is the scalar product of the reception singular vector?

16 The scalar product ( ),pw nf ce U can be interpreted as the spatial Fourier transform of the reception singular vector Un(xR), theMinonzio et al. JASA20spatial vector k being equal to 2π f/c. 

Q7. What is the Norm function for the acoustic wave?

In this case, the Norm function is given by the scalar product ( ) ( ) 2 , ,pw pw nf c f ce e equal to( ) 2R, sinc n n c cNorm f c f L c c π ⎛ ⎞−= ⎜ ⎟ ⎝ ⎠ , (10)with LR the length of the reception array. 

Q8. What is the frequency of the two probes?

The 1 MHz probe [Fig. 3(a)] presents a second harmonic bandwidth, for f ranging from 2.5 to 4 MHz, i.e. for fe ranging from 5 to 8 MHz.mm. 

Q9. What is the fe of the two probes?

Results are shown for a variable phase velocity c at a fixed frequency : f.e = 1.6 MHz.mm, with the 1 MHz probe (a) and f.e = 3.8 MHz.mm with the 2 MHz probe (b). 

Q10. What is the difference between the amplitude of a guided mode and its width?

If the amplitude of a guided mode decreases along the array, due toattenuation or geometric dispersion, the resolution function, defined by Eq. (10), will be affected in two ways: its maximum will decrease and its width will increase.