Jean-Gabriel Minonzio
email: jean-gabriel.minonzio@upmc.fr

Josquin Foiret

Maryline Talmant

Pascal Laugier

Impact of attenuation on guided mode wavenumber measurement in axial transmission on bone mimicking plates

Keywords: numbers 4320Ye Measurement methods and instrumentation, 4380Vj Acoustical medical instrumentation and measurement techniques 4320Mv Waveguides, wave propagation in tubes and ducts

Robust signal processing methods adapted to clinical measurements of guided modes are required to assess bone properties such as cortical thickness and porosity. Recently, an approach based on the singular value decomposition (SVD) of multidimensional signals recorded with an axial transmission array of emitters and receivers has been proposed for materials with negligible absorption [Minonzio et al. J. Acoust. Soc. Am. 127, 2913-2919 (2010)]. In presence of absorption, the ability to extract guided mode degrades. The objective of the present study is to extend the method to the case of absorbing media, considering attenuated plane waves (complex wavenumber). The guided mode wavenumber extraction is enhanced and the order of magnitude of the attenuation of the guided mode is estimated.

Experiments have been carried out on 2 mm thick plates in the 0.2 -2 MHz bandwidth. Two material are inspected: PMMA (isotropic with absorption) and artificial composite bones (Sawbones, Pacific Research Laboratory Inc, Vashon WA) which is a transverse isotropic absorbing medium. Bulk wave velocities and bulk attenuation have been evaluated from transmission measurements. These values were used to compute theoretical Lamb mode wavenumbers which are consistent with the experimental ones obtained with the SVD-based approach.

I. INTRODUCTION

Since more than ten years, guided waves have been known to be supported at ultrasonic frequencies by long cortical bones, such as tibia or radius. [START_REF] Moilanen | Ultrasonic guided waves in bone[END_REF][START_REF] Talmant | Bone Quantitative ultrasound Chap. 7 Guided waves in cortical bones[END_REF] The frequencydependent phase/group guided mode velocities, which are determined by bone elastic or geometrical properties, have thus the potential to be used as surrogate markers of bone strength. To date, in most in vivo axial transmission measurements, attention was focused on the measurement of the first arrival signal, denoted FAS. It has been measured on the leg (tibia) [START_REF] Foldes | Quantitative ultrasound of the tibia: A novel approach for assessment of bone status[END_REF][START_REF] Prevrhal | Quantitative ultrasound of the tibia depends on both cortical density and thickness[END_REF][START_REF] Sievanen | Ultrasound velocity and cortical bone characteristics in vivo[END_REF][START_REF] Kilappa | Lowfrequency axial ultrasound velocity correlates with bone mineral density and cortical thickness in the radius and tibia in pre-and postmenopausal women[END_REF] or the forearm (radius). [START_REF] Kilappa | Lowfrequency axial ultrasound velocity correlates with bone mineral density and cortical thickness in the radius and tibia in pre-and postmenopausal women[END_REF][START_REF] Hans | Does combining the results from multiple bone sites measured by a new quantitative ultrasound device improve discrimination of hip fracture?[END_REF][START_REF] Talmant | In Vivo Performance Evaluation of Bi-Directional Ultrasonic Axial Transmission for Cortical Bone Assessment[END_REF][START_REF] Sarvazyan | Application of the dual-frequency ultrasonometer for osteoporosis detection[END_REF] Yet, measurements of dispersion curves have not been thoroughly investigated in vivo, except in a pioneering study on tibia dedicated to A 0 mode. [START_REF] Kilappa | Lowfrequency axial ultrasound velocity correlates with bone mineral density and cortical thickness in the radius and tibia in pre-and postmenopausal women[END_REF][START_REF] Moilanen | Assessment of the tibia using ultrasonic guided waves in pubertal girls[END_REF] However, A 0 mode is more susceptible than other Lamb waves to coupling with soft tissue. [START_REF] Moilanen | Modeling the impact of soft tissue on axial transmission measurements of ultrasonic guided waves in human radius[END_REF] In vitro measurements of frequency dependent guided modes were achieved on bone specimens. Several authors used a fixed transmitter and a mobile receiver on bovine tibia.

Different frequency excitation and signal processing techniques have been considered.

Lefebvre et al. [START_REF] Lefebvre | Development of a new ultrasonic technique for bone and biomaterials in vitro characterization[END_REF] used the spatio-temporal Fourier transform at 0.1 MHz and identified three plate or Lamb modes: A 0 , S 0 and A 1 . Ta et al. [START_REF] Ta | Measurement of the Dispersion and Attenuation of Cylindrical Ultrasonic Guided Waves in Long Bone[END_REF] used the phase spectrum method with ultrasonic frequencies of 0.5 and 1 MHz. Three longitudinal tube modes were measured: L(0,1), L(0,2) and L(0,3). Le et al. [START_REF] Le | Probing long bones with ultrasonic body waves[END_REF] applied seismological principles at 1 MHz and identified different propagation paths, combination of the P and S waves. However, in all these approaches, the signals transmitted through the cortical layer along the bone axis must be recorded over a sufficiently large propagation distance (defined by the receiver scanning distance in case of a single receiver or by the length of the receiver array) in order to allow a clear separation of the different signals components in the time domain. Likewise, a large propagation distance is necessary to allow a clear separation in the frequency domain. Typical propagation distances reported in in vitro experimental works are between 60 and 160 mm, which precludes the direct application of these methods to clinical measurements. The bone segment that can reasonably be investigated is limited to approximately 20 to 30 mm due to strong bone heterogeneities and variations of soft tissue thickness along the receiving area. Time-frequency analysis with a single transmitterreceiver configuration, has been proposed to overcome the need for an extended propagation distance. [START_REF] Protopappas | Guided ultrasound wave propagation in intact and healing long bones[END_REF][START_REF] Ta | Identification and analysis of multimode guided waves in tibia cortical bone[END_REF] When applied on a sheep tibia, Protopappas et al. [START_REF] Protopappas | Guided ultrasound wave propagation in intact and healing long bones[END_REF] were able to identify several plate or Lamb guided modes A 0 , S 0 , A 1 , S 1 and A 2 . Accurate model and advanced extraction tools [START_REF] Xu | Multiridge-Based Analysis for Separating Individual Modes From Multimodal Guided Wave Signals in Long Bones[END_REF] are required to identify individual guided modes.

To overcome these limitations, a signal processing method has recently been proposed taking advantage of a multi-emitter and multi-receiver axial transmission configuration. [START_REF] Minonzio | Guided wave phase velocity measurement using multi-emitter and multi-receiver arrays in the axial transmission configuration[END_REF][START_REF] Minonzio | Measurement of guided mode wave vectors by analysis of the transfer matrix obtained with multi-emitters and multireceivers in contact[END_REF] At each frequency f of the bandwidth, the most energetic singular vectors of the response matrix are interpreted as the signal subspace basis. The so-called Norm-function corresponds to the projection into the signal subspace of non attenuated plane waves (real wavenumber k).

Guided modes are associated with Norm maxima in the (f-k) domain. The method has been validated for non dissipative materials such as copper plate, [START_REF] Minonzio | Guided wave phase velocity measurement using multi-emitter and multi-receiver arrays in the axial transmission configuration[END_REF][START_REF] Minonzio | Measurement of guided mode wave vectors by analysis of the transfer matrix obtained with multi-emitters and multireceivers in contact[END_REF] using probes initially designed for FAS clinical measurements. [START_REF] Bossy | Bidirectional axial transmission can improve accuracy and precision of ultrasonic velocity measurement in cortical bone: A validation on test materials[END_REF] In this case, measured Norm maxima values are closed to 1.

In presence of absorption, the magnitude of the maxima of the Norm-function decreases, which degrades the ability to extract guided modes. The objective of the present study is to extend the method to the case of absorbing media such as cortical bone. The Normfunction is generalized by considering attenuated plane waves (complex wavenumber k +iα).

Consequently, maxima are extracted in a three dimensional (f-k-α) domain. The Norm maxima are shown to be enhanced, allowing more reliable measurements of dispersion curves and assessment of the order of magnitude of the attenuation of the guided modes. This method is presented in Sec. II. Material and methods are then described in Sec. III. Finally experimental validation is presented in Sec. IV. Experiments are performed on polymethylacrylate (PMMA) and bone mimicking plates. Experimental complex guided mode wavenumbers are compared with theoretical predictions. Finally, for one particular case the results compared with the generalized Norm-function are compared and discussed with those obtained with the former Norm-function formalism and with the spatio-temporal Fourier transform.

II. MEASUREMENT CONFIGURATION AND SIGNAL PROCESSING

A. Axial transmission configuration

The clinical probe contains N E emitters and N R receivers, placed in the axial transmission configuration: emitters and receivers are aligned along the (Ox 3 ) axis and in contact with the wave guide to be characterized (Fig. 1). The waveguide thickness, the reception array length and the distance between the first emitters and the first receiver, are respectively denoted e, L and D. The emitter and receiver positions are denoted x j E and x j R and satisfy x i E = ip and x j R = D + jp respectively, with p the array pitch. Normal stress excitation is assumed. The length of the transducer elements along the (Ox 2 ) direction is large compared to the transducer elements width along the (Ox 3 ) direction, i.e., the array pitch. It implies that the problem can be simplified and considered as being two dimensional. An inter-element temporal response, with j and i the receiver and emitter indexes, is denoted r ji (t). Its temporal Fourier transform, denoted R ji (f), corresponds to an element of the N R ×N E response matrix R(f).

B. Response matrix

An element R ji (f) of the theoretical response matrix R(f), can be written as a sum of normal modes [START_REF] Viktorov | Rayleigh and Lamb waves[END_REF] ( )

( )( ) ( )( ) mode R E R E i 1 n j i n j i N f x x k f x x ij n n R f w e e α - - - = = ∑ , (1)
with k n (f) and α n (f) the real and imaginary parts of the guided mode wavenumber. The number of guided modes is denoted N mode at the frequency f. The modal amplitude is denoted w n and can be given, for instance, for a non absorbing plate, assuming that normal displacement is measured at the receivers by 21

( ) ( ) 2 L T = sinc , , , , . 2 2 n n nn p p w B f k E k f c c e ⎛ ⎞ ⎡ ⎤ ⎜ ⎟ ⎢ ⎥ ⎣ ⎦ ⎝ ⎠ (2)
The term B(f) corresponds to the frequency bandwidth. The sinc function corresponds to the transducer aperture function. The term E n is the intrinsic excitability of the n th mode. It can be calculated either using integral transform or reciprocity relationship. [START_REF] Viktorov | Rayleigh and Lamb waves[END_REF] Due to the two dimensional assumption, no geometric dispersion is considered as the propagation is purely in (Ox 1 x 3 ) plane. Thus the amplitude decay is due to the guided mode attenuation and varies as

( ) 3 n f x e α -
. Transducers of finite dimension along the (Ox 2 ) axis have been described in Ref. [START_REF] Moulin | Pseudo-3D modeling of a surfacebonded Lamb wave source (L)[END_REF] and are not considered here.

Considering the two previous equations, the theoretical response matrix can be written

R(f) = a R W t a E* .
The notation t and * correspond to the transpose and complex conjugation operations. Elements of the reception and emission matrices a R and a E are

( ) ( ) ( ) R R exp i jn n n j a f k f x α = - + ⎡ ⎤ ⎣ ⎦ , ( ) ( ) (
)

E E exp i . in n n i a f k f x α = + ⎡ ⎤ ⎣ ⎦ (3)
The matrix W is diagonal and contains the weighting terms w n [Eq.( 2)], i.e., W nm = w n δ nm with δ nm the Kronecker symbol. The theoretical response matrix can be expressed using normalized vectors as

R(f) = e R S t e E* . (4)
The normalized guided mode vectors in emission or reception satisfy

E,R E,R E,R . n n n = a e a (5)
The notation ||a E n || corresponds to the norm of the a E n vector given by Eq. ( 3). Its expression is given in Eq. (A3) of the appendix. With N mode the number of guided modes at the frequency f, the dimensions of the reception and emission normalized guided mode matrices e R and e E are N R × N mode and N E × N mode respectively. The N mode × N mode matrix S is diagonal and contains the normalized weighting terms

S n = w n ||a R n || ||a E n ||. (6)

C. Norm-function expression

In a first step, the singular value decomposition (SVD) is applied to the response matrix R(f). The SVD of the response matrix writes as a three matrix product

R (f) = U Σ t V * . ( 7
)
The matrices U and V contain the reception and emission singular vectors. ). The singular vectors associated with the signal subspace, i.e., U m≤M , form the received signal basis.

Consider an attenuated spatial plane wave with a complex wavenumber k+iα, denoted e test (k, α). It is defined as the test normalized guided mode following Eqs. ( 3) and ( 5) as

e test (k, α) = e R (k, α). (8)
The test vector e test (k, α) is expressed in the canonical basis , i.e., the receiver basis [Eq. ( 3)].

It can be projected in the signal subspace, using the singular vector basis, as

1 . M test test signal m m m= = ∑ e U e U (9)
The Hermitian scalar product <U m | e test > is equal to t U m * . e test . The Norm-function is defined at each frequency, as the square of the norm of the previous expression

( ) ( ) ( ) 2 1
, , , .

M test m m Norm f k f k α α = = ∑ U e ( 10
)
The Norm-function is real positive. Due to the normalization [Eqs. ( 5) and ( 7)], it ranges from 0 to 1. Equation ( 10) corresponds to the extension of the non attenuated case. [START_REF] Minonzio | Guided wave phase velocity measurement using multi-emitter and multi-receiver arrays in the axial transmission configuration[END_REF] In this previous formalism, the Norm-function was expressed in the two dimensional (f, k) plane.

Here, the Norm-function is expressed in the three dimensional (f, k, α) space, where α is the imaginary part of the testing wavenumber. Thus, the previous definition is included in the new formalism and corresponds to the α = 0 line of a (k, α) plane.

D. Norm-function interpretation

The Norm-function can be interpreted using the guided mode basis. Equations ( 4) and ( 7) are formally similar, i.e., the response matrix R(f) can be expressed in function of the normalized guided modes basis (e n R ) or in function of the signal singular vectors basis (U m≤M ).

Due to the finite reception array length, the normalized guided modes e n R are not orthogonal.

On the contrary, the singular vectors U m≤M are defined orthogonal. [START_REF] Hua | Matrix Pencil Method for Estimating Parameters of Exponentially Damped Undamped Sinusoids in Noise[END_REF] The projection in the signal subspace of the test vector e test can be expressed using the singular vector basis [Eq. ( 9)] or using the normalized guided mode basis as

( ) mode R R 1 , . N test test signal n n n k α = ≈ ∑ e ee e (11)
If k +iα is close to a guided mode wavenumber k n +iα n , the test vector e test (k, α) is projected mainly on the n th guided mode ( )

R , n n n k α e as ( ) ( ) ( ) R R , , , ,
test test n n signal n n n n n n k k k k α α α α ≈ e e e e ∼ ∼ (12)
and the Norm-function [Eq. (10)] is given in that case by

( ) ( ) ( ) 2 R , , , , . test n n n n n Norm f k k k k α α α α = e e ∼ ∼ (13)
The expression of the Hermitian scalar product does not correspond to a guided mode wavenumber, the Norm-function is small compared to 1. The resolution, i.e., the ability to discriminate two close modes, corresponds to the main lobe width. [START_REF] Minonzio | Guided wave phase velocity measurement using multi-emitter and multi-receiver arrays in the axial transmission configuration[END_REF] Thus the real and imaginary part of the guided mode wavenumbers can be obtained from the Norm maxima positions in the (f, k, α) 3D-space. A second threshold t 2 is used to select the retained maxima larger than t 2 . As the maximum value reflects how the spatial plane wave is represented in the basis of the signal subspace, the value is interpreted as a quality parameter ranging from 0 to 1.

As discussed in

III. MATERIAL AND METHODS

A. Samples

Two Panametrics V152 RM) contact transducers [START_REF] Granke | Change in porosity is the major determinant of the variation of cortical bone elasticity at the millimeter scale in aged women[END_REF] using a conventional through-transmission method on 10 mm-thick plate of the same material. [START_REF] Sachse | Determination of Phase and Group Velocities of Dispersive Waves in Solids[END_REF][START_REF] Chaffai | Ultrasonic characterization of human cancellous bone using transmission and backscatter measurements: Relationships to density and microstructure[END_REF][START_REF] Peters | A broad band spectroscopy method for ultrasound wave velocity and attenuation measurement in dispersive media[END_REF] Constant rate of attenuation increase with frequency are experimentally observed. [START_REF] Szabo | Causal theories and data for acoustic attenuation obeying a frequency power law[END_REF] Thus, the bulk wave absorption per wavelength (αλ) L,T is constant, where subscripts L and T refer to compression and shear waves respectively. With this notation, a bulk plane wave can be written as

exp[ ik L,T (1 + i(αλ) L,T /2π)x ].
The measured values compression (c L ) and shear (c T ) bulk velocities and compression (αλ) L and shear (αλ) T attenuation, the density ρ and the plate thickness e are reported in Table I. The PMMA values are consistent with previously reported values. [START_REF] Treiber | Correction for partial reflection in ultrasonic attenuation measurements using contact transducers[END_REF][START_REF] Hartmann | Ultrasonic Hysteresis Absorption in Polymers[END_REF][START_REF] Hosten | Reflection and transmission coefficients for guided waves reflected by defects in viscoelastic material plates[END_REF][START_REF] Longo | Simultaneous Determination of Acoustic Velocity and Density of a Cortical Bone Slab: Ultrasonic Model-Based Approach[END_REF] For the bone mimicking material, the bulk velocities and attenuation are first measured perpendicular (⊥) to the fibers using the 10 mm-thick plate. In order to evaluate the longitudinal properties parallel (||) to the fibers, measurement have been done on a small cube (about 10×10×10.

B. Lamb wave calculations

Theoretical Rayleigh-Lamb spectrum was calculated for isotropic (PMMA) and transverse isotropic plates (bone mimicking material). [START_REF] Talmant | Bone Quantitative ultrasound Chap. 7 Guided waves in cortical bones[END_REF][START_REF] Coquin | Attenuation of Guided Waves in Isotropic Viscoelastic Materials[END_REF][START_REF] Chan | Lamb waves in highly attenuative plastic plates[END_REF] For the bone mimicking plate, two cases are considered, depending if the probe is placed perpendicularly or parallel to the fibers. In experiments, as the probe works in piston mode and does not excite horizontal shear displacement, only motion in the plane (Ox 1 x 3 ) is considered. [START_REF] Rhee | The group velocity variation of Lamb wave in fiber reinforced composite plate[END_REF] Due to geometrical hypothesis discussed in paragraph II.A, the perpendicular orientation is similar to an isotropic case. Thus, in this case, an absorbing plate with linear variation of absorption with frequency was assumed. For small values of (αλ) L,T , the real part of the guided mode wavenumbers is close to the non absorbing values. If the probe is placed parallel to the fibers, transverse isotropic case is considered. Nevertheless, as the attenuation could not be measured for all stiffness coefficients (Table I), only the non absorbing transverse isotropic is considered. The c 13 coefficient was deduced from c 33 and c 44 using c 13 = c 33 -2c 44 .

C. Axial transmission measurements

The custom made probe used in this study contains 32 receivers and 5 emitters. [START_REF] Minonzio | Measurement of guided mode wave vectors by analysis of the transfer matrix obtained with multi-emitters and multireceivers in contact[END_REF] The reception array pitch, denoted p, is equal to 0.

IV. EXPERIMENTAL RESULTS

A. Guided mode wavenumbers of the PMMA plate

First, axial transmission measurements have been carried out on the isotropic dissipative PMMA plate. The plate thickness e measured with a caliper was 1.95 mm (Table I). The real and imaginary parts of wavenumbers k n I). Modes vicinity may also affect attenuation evaluation as for example S 1 and A 2 around 1.7 MHz. The two modes with longitudinal asymptotic behavior (S 1 and A 2 ) are experimentally more attenuated than expected from the Lamb free plate spectrum. It may be due to the coupling between the probe and the wave guide which has not been taken into account in the model.

B. Guided mode wavenumbers of the bone mimicking plate

The bone mimicking plate thickness has been next measured to 2.3 mm with a caliper (Table I). First, the probe is placed perpendicularly to the fibers. Due to the geometrical hypothesis described in Sections. II and III, this configuration can be considered as isotropic.

Thus, an analysis similar to the PMMA case has been used. The same threshold t 2 = 0.85 has been used. The results are shown in Fig. 5 with the theoretical values. The same modes are observed as in the PMMA case, but they appear in a different order, because of the value taken by the ratio c L /2c T , which is larger than 1 for PMMA and less than 1 for the bone mimicking plate measured perpendicularly to the fibers (Table I). The agreement between experimental and theoretical data is overall good and of comparable to the PMMA case, except for the mode S 2 around 1.2 MHz and for the modes

C. Discussion

In order to illustrate the method, the particular case of the PMMA plate at 1.84 MHz is now discussed. This case is marked with a vertical dash-dot line on Fig. 3. At this frequency, three guided mode are measured: A 5 , A 2 and S 1 and the experimental rank (or signal subspace dimension) M is equal to 3. The corresponding Norm(f, k, α) function [Eq. ( 10)], i.e.,

|<U 1 | e test (k,α)>| 2 + |<U 2 | e test (k,α)>| 2 + |<U 3 | e test (k,α)>| 2
, is illustrated on Fig. 7(a) in the (k, α)

plane with f being equal to 1.84 MHz. Three peaks (shown with crosses) are observed in the (k, α) plane corresponding to the experimental wavenumbers. The real part of the wavenumber k n exp (f) are equal to 1.9 (A 5 ), 3.7 (A 2 ) and 4.6 rad.mm -1 (S 1 ). The corresponding imaginary part α n exp (f) equal to 0.15 (A 2 ) 0.2 (S 1 ) and 0.4 Np.mm -1 (A 5 ). In this example, the peak values for these three modes, range from 0.9 to 0.95. Thus the second threshold t 2 , chosen equal to 0.85, allows to select to the peaks associated with these guided modes. This threshold is shown with a thick line on Fig. 7(a).

As attenuation increases, the useful reception length decreases. Figure 8 shows the modulus of different normalized guided modes e n R (k n ,α n ) with α n ranging from 0 to 0.5 rad.mm -1 . For example, with α n equal to 0.4 Np.mm -1 , the measured amplitude at the middle of the reception array is -40 dB below the amplitude of the signal measured at the first receiver. In this case, only half of the reception array can be considered as useful, the second half of received signals being with poor signal-to-noise ratio. It implies that the main lobe width increases as the attenuation increases [Eq. (A6) and Fig. 7]. The peak associated with A 2 is larger than expected due to the vicinity of the S 4 mode. The two modes are too close to be resolved. In this paper, the authors specifically addressed the influence of attenuation on the measurement of guided mode wavenumbers in a bone mimicking plate. The material, is a homogenous absorbing transverse isotropic material which is considered to reasonably match the anisotropic elastic properties of actual human cortical bones. [START_REF] Granke | Change in porosity is the major determinant of the variation of cortical bone elasticity at the millimeter scale in aged women[END_REF] The disadvantage of performing measurements on phantoms, of course, is that they only approximate true cortical bone structure. In order to generalize the approach to the in vivo measurement conditions, several parameters will have to be taken into account in future studies, such as the irregular geometry, the heterogeneity of the cortical tissue and the presence of marrow and surrounding soft tissues. Performing the measurements with a relatively short reception length of ~20-30 mm potentially minimizes the adverse impact of irregular geometry and of bone heterogeneity. The presence of surrounding tissues and marrow will likely affect the wavenumber measurement. However, our preliminary (data not shown) obtained in a bone mimicking plate covered by a soft-tissue mimicking layer suggested that the guided mode wavenumbers are only slightly modified compared to their counterparts in the free bone mimicking plate, and that they are still well predicted by an elastic transverse isotropic waveguide. The authors' intent is to progressively extend the study first to the case of bone phantoms with soft tissue mimicking layers on top of it and marrow inside and then to in vitro experiments with real inhomogeneous bone specimens. These critical issues will be the object of our future studies in order to prepare the in vivo application.

Conclusion

An original method for evaluating the complex guided mode wavenumbers, using a probe consisting of two separated emission and reception arrays in contact with the inspected waveguide, is presented in this paper. The method is a generalization of the Norm-function formalism described previously for non attenuated guided mode. This method allows the evaluation of both real and imaginary parts of the attenuated guided mode wavenumbers.

Experimental results are in good agreement with the theoretical Lamb modes. The methodology presented herein hold promises for the assessment of the structural and material properties of wave guides, and thus can potentially be used for assessing cortical bone strength. Future studies will focus on the evaluation of elastic properties of cortical bone and will be extended to in vivo measurements. The impact of the geometry (e.g., variable cortical thickness) and soft tissue coupling will also be taken into account.

1 , , R n j n j N k k x x test n n n pw j n k k e α α α α ⎡ ⎤ - - + ⎣ ⎦ = = ∑ e e a a . (A4)
In the general case, i.e., k ≠ k n and α ≠ α n , the scalar product is complex. The analytical solution can also be evaluated, using an integral solution

( ) ( ) ( ) i R R 0 1 , , L k x test n n n pw n dx k k e p α α α Δ -Δ = ∫ e e a a , (A5)
with Δk = k n -k and Δα = α n + α. The solution of the previous equation is The previous Norm-function formalism corresponds to the α = 0 case and its maximum also corresponds to k equal to k n . [START_REF] Minonzio | Guided wave phase velocity measurement using multi-emitter and multi-receiver arrays in the axial transmission configuration[END_REF] The value of this maximum depends on the α n L product as

( ) ( ) ( )( ) ( ) i R 2 2 1 , , 2 i 1 1 n k L test n n n n L L e k k k e e α α α α α α α α Δ -Δ - - - = Δ -Δ - - e e . ( A6
( ) ( ) ( ) 2

FIG. 1 .

 1 FIG. 1. Geometry of the problem: emission and reception arrays are placed in the axial

1 RFIG. 2 . 1 R , e 2 R)

 1212 FIG. 2. Schematic representation of a signal subspace of rank 2: the test vector e test can be

  Eq. (A6) in the appendix. This scalar product can be interpreted as the normalized spatial Fourier transform of the normalized guided mode ( ) , the SVD can be interpreted as a denoising step included between the temporal and spatial Fourier transforms. If k +iα corresponds to a guided mode wavenumber k n +iα n , the Norm-function is close to 1. If k +iα

  the paragraph II.C, the definition of the Norm-function adapted to the case of non-attenuating propagation media corresponds to the α = 0 2D-restriction of the (f, k, α) 3D-space. The 2D point of view is limited in case of absorbing waveguides, because the maxima of the Norm(f, k, 0) function decreases as α n increases [Eq. (A7)]. Thus maxima can be of the order of other mode secondary maxima or simply associated with noise. Thus, in presence of absorption, the magnitude of the maxima of the 2D-Norm-function decreases, this degrades the ability to extract guided mode. On the contrary, with the 3D-Norm-function definition, the maxima are enhanced and closer to 1. Thus a second threshold t 2 closer to 1 can be chosen. Moreover, the order of magnitude of the attenuation of the guided mode a n (f) is estimated. An example is shown and discussed in Sec. IV.C.

  8 mm. The reception array length, denoted L, is about 26 mm. The probe is driven by a custom made multichannel emission and reception electronic system (Althaïs Technologie, Tours, France). Emitted signals are wideband pulses (-50 dB bandwidth 0.2-2 MHz). The sampling frequency is equal to 20 MHz. The number of recorded time samples is equal to 1024. Fifty times signals are averaged to increase signal-tonoise ratio.The experimental guided mode wavenumbers are extracted from the Norm-function in the following way. In a first step, the inter-element responses r ji (t) are experimentally recorded. Then, the experimental response matrix R(f) is computed using the temporal Fourier transform for each frequency of the bandwidth. The SVD is then computed at each frequency and the heuristic threshold t 1 is applied on the singular values in order to define the signal subspace dimension M, as discussed in paragraph II.C. Finally, the Norm-function is computed in the (f, k, α) 3D-space following Eq.(10). The experimental wavenumbers are obtained from the values of the Norm-function larger than a second threshold t 2 equal here to 0.85. Experimental results are shown and discussed in Sec. IV. For one particular case, the former Norm-function, i.e., for α = 0, and the spatio-temporal Fourier transform are also performed. These results are compared with those obtained the generalized Norm-function in this particular case.

FIG. 3 .FIG. 4 .

 34 FIG. 3. Results for the PMMA plate: real (a) and imaginary parts (b) of the guided mode

  (a) and (b). The experimental real k n exp (f) and imaginary part α n exp (f) of the guided mode wavenumber (shown with symbols) are compared

A 2 FIG. 5 .FIG. 6 .

 256 FIG. 5. Results for the 2.3 mm bone mimicking plate, measured with the probe oriented

FIG. 7 .FIG. 8 .

 78 FIG. 7. Norm(f, k, α) function in the (k, α) plane for f = 1.84 MHz (a) and for α = 0 (b),

)

  For quasi orthogonal guided modes, the Norm-function is equal to |<e n R |e test >| 2 [Eq. (13)]. For k = k n and α =α n , the scalar product and the Norm-function are maximum and equal to 1. The Norm-function decrease and lead toward zero as Δk and Δα increase. With α n and α nulls, the scalar product <e n R |e test > is equal to

  equation allows the estimation imaginary part α n from the maximum values of the 2D-Norm(f, k)-function.[START_REF] Minonzio | Guided wave phase velocity measurement in anisotropic absorbing material[END_REF]

FIG. 3 .FIG. 4 .FIG. 5 . 35 FIG. 7 .FIG. 8 .< U 1 | e test > < e 2 R| e t e s t > signal < e 1 R

 3453578121 FIG. 3. Results for the PMMA plate: real (a) and imaginary parts (b) of the guided mode

  materials have been measured: PMMA and bone mimicking plates. The plate thicknesses, about 2 mm, correspond to a typical cortical bone thickness. The PMMA is considered as isotropic. The bone mimicking material (Sawbones, Pacific Research Laboratory Inc, Vashon WA) is made of oriented glass fibers mixed with epoxy. The material

is transverse isotropic, with elastic properties close to those of real bones. The velocities and attenuation coefficients of bulk compression and shear waves of both materials were determined with longitudinal (2.25 MHz Panametrics V105RM) and transverse

(1 MHz

TABLE I .

 I The measured values are consistent with previously reported values on Sawbones 33 or epoxy plates.34 These values are used to compute theoretical guided mode wavenumbers as discussed in paragraph III.B. Theoretical predictions are then compared to experimental data in Sec. IV. Measured values for PMMA and bone mimicking (bone mim.) plates

		c L (m.s -1 ) c T (m.s -1 )	(αλ) L (Np)	(αλ) T (Np)	ρ (g.cm -3 ) e (mm)
	PMMA	2685 ± 20 1365 ± 10 0.059 ± 0.007 0.063 ± 0.003	1.18	1.95
	bone mim. ⊥ 2870 ± 40 1520 ± 20 0.087 ± 0.013 0.147 ± 0.004	1.64	2.30
	bone mim. ||	3705 ± 40 1585 ± 20

2 mm 3 ) extracted from the plate. The cube dimensions were large enough to evaluate the parallel bulk velocities c L || and c T || but too small to evaluate the attenuations (αλ) L || and (αλ) T || .

TABLE I .

 I Measured values for PMMA and bone mimicking (bone mim.) plates

	c L (m.s -1 ) c T (m.s -1 ) 2685 ± 20 1365 ± 10 0.059 ± 0.007 0.063 ± 0.003 (αλ) L (Np) (αλ) T (Np) bone mim. ⊥ 2870 ± 40 1520 ± 20 0.087 ± 0.013 0.147 ± 0.004 PMMA transmission configuration. FIG. 2. Schematic representation of a signal subspace of rank 2: the test vector e test can be ρ (g.cm -3 ) e (mm) 1.18 1.95 1.64 2.30 projected on the singular vector basis (U 1 , U 2 ) [Eq. (9)] or in normalized guided mode basis bone mim. || FIG. 1. Geometry of the problem: emission and reception arrays are placed in the axial (e 1 R , e 2 R ) [Eq. (11)].

Acknowledgments

Authors would like to acknowledge Pierre Nauleau for his help for the transverse transmission measurements and Mohamed Cherrad for the axial transmission measurements. This work has been supported by ANR project "COSTUM" 09-TECS-005-03 (2009-2012).

APPENDIX

In order to lighten the theoretical part (Sec. II), some notations used are explicitly calculated in this appendix. In particular, the expressions of the norm of the guided mode vector a E n and the scalar product <e n R | e test > are given here. Consider a single attenuated guided mode a n R given in Eq. ( 3). The norm of the

a a e

The analytical solution can be evaluated, using an integral solution

with p, the array inter element pitch. Thus, the norm is

If α n L tends toward zero, the normalization factor tends toward R N as in Ref. [START_REF] Minonzio | Guided wave phase velocity measurement using multi-emitter and multi-receiver arrays in the axial transmission configuration[END_REF] .

The scalar product between the normalized guided mode e n R [Eq. ( 5)] and the test vector e test [Eq. ( 8)]


Jean-Gabriel Minonzio
email: jean-gabriel.minonzio@upmc.fr

Marilyne Talmant

Pascal Laugier

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

Keywords: numbers 4320Ye Measurement methods and instrumentation, 4380Vj Acoustical medical instrumentation and measurement techniques 4320Mv Waveguides, wave propagation

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

Jean-Gabriel Minonzio, Marilyne Talmant, Pascal Laugier

I. INTRODUCTION

Different quantitative ultrasound (QUS) techniques are currently developed for clinical assessment of human bone status. [START_REF] Laugier | Instrumentation for in vivo assessment of bone strength[END_REF][START_REF] Gluer | A new quality of bone ultrasound research[END_REF][START_REF] Laugier | Quantitative ultrasound of bone: looking ahead[END_REF] The goal is to determine ultrasound-based indicators of bone strength, suitable for the discrimination of osteoporotic patients from healthy subjects. [START_REF] Krieg | Quantitative ultrasound in the management of osteoporosis: the 2007 ISCD Official Positions[END_REF] Alternatively, the possibility of bone properties determination from ultrasound measurements, for instance cortical thickness or elasticity, has been explored. [START_REF] Moilanen | Assessment of the cortical bone thickness using ultrasonic guided waves: Modelling and in vitro study[END_REF][START_REF] Lefebvre | Development of a new ultrasonic technique for bone and biomaterials in vitro characterization[END_REF][START_REF] Desceliers | Determination of the random anisotropic elasticity layer using transient wave propagation in a fluid-solid multilayer : model and experiments[END_REF] 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. [START_REF] Minnock | Using Risk Factors and Quantitative Ultrasound to Identify Postmenopausal Caucasian Women at Risk of Osteoporosis[END_REF][START_REF] Maata | Correlation of Tibial Low-Frequency Ultrasound Velocity with Femoral Radiographic Measurements and BMD in Elderly Women[END_REF][START_REF] Sarvazyan | Application of the dual-frequency ultrasonometer for osteoporosis detection[END_REF] 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. [START_REF] Moilanen | Ultrasonic guided waves in bone[END_REF] 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. [START_REF] Barkmann | A new method for quantitative ultrasound measurements at multiple skeletal sites -First Results of precision and fracture discrimination[END_REF][START_REF] Bossy | Bidirectional axial transmission can improve accuracy and precision of ultrasonic velocity measurement in cortical bone: a validation on test materials[END_REF] The FAS can be interpreted as a guided S 0 wave in the low frequency regime (i.e., low cortical thickness-towavelength ratio) and as a lateral compression wave in the high frequency regime (i.e., high cortical thickness-to-wavelength ratio). [START_REF] Bossy | Effect of bone cortical thickness on velocity measurements using ultrasonic axial transmission: A 2D simulation study[END_REF] 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 [START_REF] Moilanen | Measuring guided waves in long bones: Modeling and experiments in free and immersed plates[END_REF][START_REF] Alleyne | A two-dimensional Fourier transform method for the measurement of propagating multimode signals[END_REF] or the singular value decomposition in the space-time domain. [START_REF] Sasso | Analysis of the most energetic late arrival in axially transmitted signals in cortical bone[END_REF][START_REF] Vrabie | Singular value decomposition by means of independent component analysis[END_REF] 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 velocities. The L(0, n) longitudinal tube wave phase velocities (1 ≤ n ≤ 3) were measured on a bovine tibia. [START_REF] Ta | Measurement of the dispersion and attenuation of cylindrical ultrasonic guided waves in long bone[END_REF] Authors used angled beam transducers and analyzed the received signals with the phase spectrum method. [START_REF] Sachse | On the determination of phase and group velocities of dispersive waves in solids[END_REF] 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. [START_REF] Protopappas | Guided ultrasound wave propagation in intact and healing long bones[END_REF] Authors used two fixed transducers and obtained group velocities using time-frequency analysis. [START_REF] Prosser | Time-frequency analysis of the dispersion of Lamb modes[END_REF] 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. [START_REF] Hans | Hip fracture discrimination study -QUS of the radius and the calcaneum[END_REF][START_REF] Knapp | Multisite quantitative ultrasound: Precision, age-and menopause-related changes, fracture discrimination, and T-score equivalence with dual-energy X-ray absorptiometry[END_REF][START_REF] Talmant | In vivo performance evaluation of bi-directional ultrasonic axial transmission for cortical bone assessment[END_REF] In addition, a FAS velocity database, measured in vivo, was used in a technique of elasticity identification. [START_REF] Desceliers | Determination of the random anisotropic elasticity layer using transient wave propagation in a fluid-solid multilayer : model and experiments[END_REF] From A 0 phase velocity data, inversion schemes were proposed to estimate cortical thickness. [START_REF] Moilanen | Assessment of the cortical bone thickness using ultrasonic guided waves: Modelling and in vitro study[END_REF] 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. [START_REF] Moilanen | Modeling the impact of soft tissue on axial transmission measurements of ultrasonic guided waves in human radius[END_REF] 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 [START_REF] Alleyne | A two-dimensional Fourier transform method for the measurement of propagating multimode signals[END_REF][START_REF] Vrabie | Singular value decomposition by means of independent component analysis[END_REF][START_REF] Sachse | On the determination of phase and group velocities of dispersive waves in solids[END_REF][START_REF] Prosser | Time-frequency analysis of the dispersion of Lamb modes[END_REF] 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 singular vectors and singular values is adapted and specific to each case and array configuration. [START_REF] Kolda | Tensor Decompositions and Applications[END_REF][START_REF] Neilsen | Extraction of acoustic normal mode depth functions using vertical line array data[END_REF][START_REF] Cheney | The linear sampling method and the MUSIC algorithm[END_REF] The paper is organized as follows. First, the principle of the method is presented in Sec. II: the phase velocities are measured using the singular vectors of the response matrix, interpreted as the basis of the received signal subspace. The advantages of the method are then described: the signal and noise separation and the improvement of the ability to evaluate phase velocities. To this end, two specific thresholds are discussed. In Sec. III the accuracy and effectiveness of the method are tested on experimental measurements on a reference wave guide. Two probes of central frequency equal to 1 and 2 MHz, developed for the axial transmission approach, are used. [START_REF] Bossy | Bidirectional axial transmission can improve accuracy and precision of ultrasonic velocity measurement in cortical bone: a validation on test materials[END_REF] The experimental set up, the evaluation domain and the resolution are then discussed. Finally experimental phase velocities arecompared with theoretical ones and with those obtained with the spatio-temporal Fourier transform.

II. THEORY

A. Response matrix

A single probe consisting of two separated arrays are used in contact with a metallic plate, using coupling gel (Fig. 1). Superscripts E and R refer respectively to the emission and reception arrays. Thus the emission array contains N E emitters of position

x i E , with 1≤ i ≤ N E .
Likewise, the reception array contains N R receivers of position x j R , with 1≤ j ≤ N R . The response to a pulse emitted by the ith emitter, received by the jth receiver after propagation in the medium, is denoted h ji (t). It corresponds to the impulse response between the two transducers. The number of responses is equal to N R multiplied by N E . The received signal r ji (t) on the jth receiver is given by the sum of the convolutions of the emissions e ji (t)

with the impulse responses h ji (t) as

( ) ( ) ( ) E 1 N ji ji ii i r t h t e t = = ⊗ ∑ . ( 1
)
The temporal Fourier transform of Eq. ( 1) writes

( ) ( ) ( ) E 1 N ji ji ii i R f H f E ω = = ∑ , (2)
with f the frequency and R j (f), H ji (f), E i (f) the temporal Fourier transforms of r j (t), h ji (t), e i (t)

respectively. The N R ×N E matrix R contains the N R ×N E Fourier transforms R ji (f). Likewise, the

N E × N E diagonal matrix E contains the N E Fourier transforms E ii (f).
The relation between the two vectors R and E writes, at a given frequency f

R = H. E , (3)
with H the N R ×N E matrix containing the elements H ji (f) equal to the Fourier transforms of the impulse responses h ji (t). The R matrix, called afterward "response matrix", is characteristic of the "emission -propagation medium -reception" system (Fig. 1).

B. Singular vectors basis

The Singular Value Decomposition (SVD) of the response matrix R writes

E * 1 N t n n n n σ = = ∑ R U V , (4)
where the notations t and * denote the transpose and conjugation operations. The number of experimental singular values σ n is equal to minimum size of the arrays, i.e. the minimum between N E and N R , equal in the following to N E . The notation V n refers to an emission singular vector, of dimension N E ×1, and U n to a reception singular vector, of dimension N R ×1.

These two vectors are associated with the singular value σ n . The reception singular vectors U n are the eigenvectors of R. t R * , whereas, the emission vectors V n are the eigenvectors of t R * . R. In both cases the eigenvalues are the square of the singular values σ n . [START_REF] Prada | Separation of interfering acoustic scattered signals using the invariants of the time-reversal operator. Application to Lamb waves characterization[END_REF] In the following, the singular values are considered as sorted in a decreasing order, i.e.

σ 1 > σ 2 > …
The singular vectors U n (resp. V n ) are normalized and define an orthogonal basis of the received (resp. emitted) signals. In particular any spatial plane wave, propagating along the reception array direction, i.e. along the x axis (Fig. 1), can be expressed in the reception basis.

Consider a spatial plane wave e pw (f, c) depending on the phase velocity c and the frequency f, defined on the jth receiver as

( ) R R 1 2 , e x p i pw j j f e f c x c N π ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ , (5)
The previous vector is normalized. As the phase velocity c is real, equation 5 corresponds to the propagating part of the spatial plane wave. Its expression on the basis defined by the reception singular vectors U n is

( ) ( ) E 1 , , N pw pw n n n f c f c = = ∑ e e U U, (6)
with the notation ( )

, pw n
f c e U corresponding to the Hermitian scalar product, equal to ( ) * , .

t pw n f c ⎡ ⎤ ⎣ ⎦ e U .
If the array is placed along the (Oy) direction (Fig. 1), as for example a vertical array in shallow water guide, [START_REF] Neilsen | Extraction of acoustic normal mode depth functions using vertical line array data[END_REF] each reception singular vector U n is associated with a single guided mode. If the array is distant from the wave guide in the DORT configuration, DORT being the French acronym for decomposition of the time-reversal operator, the singular vectors are associated with guided mode scattering directions. [START_REF] Prada | Separation of interfering acoustic scattered signals using the invariants of the time-reversal operator. Application to Lamb waves characterization[END_REF] In the axial transmission geometry, as the received signals are measured along the (Ox) direction, a reception singular vector corresponds to a combination of guided modes as discussed in paragraph III.D.

C. Noise and signal separation

One of the advantages of the SVD approach is its ability to separate noise and signal subspaces. [START_REF] Cheney | The linear sampling method and the MUSIC algorithm[END_REF][START_REF] Prada | Separation of interfering acoustic scattered signals using the invariants of the time-reversal operator. Application to Lamb waves characterization[END_REF][START_REF] Davy | Influence of noise on subwavelength imaging of two close scatterers using time reversal method: Theory and Experiments[END_REF] Indeed, using an intermediate order m corresponding to the limit between the two subspaces, Eq. ( 6) can be rewritten as

( ) ( ) ( ) E 1 1 , , , m N pw pw pw n n n n n n m f c f c f c = = + = + ∑ ∑ e e U U e U U. (7)
The order m is defined at each frequency using a threshold t 1 applied to the singular values σ n .

On the one hand, if the singular value is larger than that the threshold t 1 , the corresponding reception singular vector U n is associated with the signal subspace. On the other hand, if the singular value is less than that the threshold t 1 , the corresponding reception singular vector U n is associated with noise. In the following, the signal singular vectors are retained whereas the noise singular vectors are eliminated. Thus the norm of the propagating spatial plane wave on the basis of the signal subspace becomes

( ) { } ( ) 2 1
, ,

n m m pw pw n n f c f c ≤ = = ∑ U e e U , (8)
with the notation | | used for the modulus of complex numbers.

The value of the threshold t 1 is heuristically chosen, based on an estimation of the signal-to-noise ratio. It depends on the estimated number of measurable guided modes and on the noise level at each frequency as discussed in Ref. 31. The threshold can be chosen fixed or variable with frequency and attenuation. Examples are discussed in Sec. III.

D. Evaluation of the guided mode phase velocities

As m, the dimension of the signal subspace, is less than the number of receivers N R , i.e. the dimension of the vector e pw (f, c) [Eq. ( 5)], the basis defined by the signal reception singular vectors U n≤m is incomplete. It implies that the norm of the propagating spatial plane wave expressed on the basis of the signal subspace, i.e.

( ) { }

, (8)], is less than 1. Consequently, each value in the (f, c) domain of that norm ranges from 0 to 1. This value reflects how the spatial plane wave is represented in the basis of the signal subspace. On the one hand, if the value is small compared to 1, the spatial plane wave is absent of the received signals. On the other hand, if the value is close to 1, the spatial plane wave corresponds to a experimental propagating guided mode, present in the received signals. To enhance the contrast between the low and high values, the square of the norm is used in the following.

n m pw f c ≤ U e [Eq.
And a second threshold t 2 is then applied to the norm in order to reduce the range from t 2 to 1.

Thus a new function, denoted Norm(f, c), is defined as

( ) ( ) { } ( ) { } ( ) ( ) { } 2 2 2 2 2 , , , , 0 , n m n m n m pw pw pw Norm f c f c if f c t Norm f c if f c t ≤ ≤ ≤ ⎧ = ≥ ⎪ ⎨ ⎪ = < ⎩ U U U e e e . (9)
A maximum of the Norm function in the (f, c) domain, larger than t 2 , provides the phase velocity c of a guided mode present in the signal subspace at the frequency f. The value of the second threshold t 2 is also heuristically chosen considering the variations of the square of the norm

( ) { } 2 , n m pw f c ≤ U e .
In the examples discussed in Sec. III the threshold t 2 is chosen equal to 0.5.

III. EXPERIMENTAL RESULTS AND DISCUSSION

A. Experimental set up

Two probes developed at the LIP (Laboratoire d'Imagerie Paramétrique), for the bone characterization, are used. [START_REF] Bossy | Bidirectional axial transmission can improve accuracy and precision of ultrasonic velocity measurement in cortical bone: a validation on test materials[END_REF] Their central frequencies f 0 are equal respectively to 1 and 2

MHz. The number N E of emitters is equal to 3 (f 0 = 1 MHz) and 8 (f 0 = 2 MHz). In both cases, the number of receivers N R is equal to 14. Thus the number of emitters, which is less than the number of receivers, corresponds to the number of experimental singular values. A coupling gel is used in order to improve the acoustical impedance matching. Emitted signals are pulses with bandwidth centred on the central frequency f 0 . The reception array pitch is equal to 0.80 mm (f 0 = 1 MHz) and 0.38 mm (f 0 = 2 MHz).

The experimental scattering medium is a 2 mm thickness copper plate. The free plate guided modes are given by the Rayleigh-Lamb dispersion equation. [START_REF] Royer | Elastics waves in solids 1 : Free and guided propagation[END_REF] No attenuation is considered in this model. The modes are numerated following their cut-off frequencies f c . [START_REF] Royer | Elastics waves in solids 1 : Free and guided propagation[END_REF][START_REF] Prada | Local vibration of an elastic plate and zero-group velocity Lamb modes[END_REF] The cut-off frequencies of the A 2n+1 and A 2n modes satisfy respectively f c e = (n+½)c T and nc L , with e the plate thickness (Fig. 1). Likewise, the cut-off frequencies of the S 2n+1 and S 2n modes satisfy respectively f c e = (n + ½)c L and nc T . These frequencies are shown in the upper scale of Fig. 4.

B. Resolution and evaluation domain of the system

The resolution is defined similarly to the spatio-temporal Fourier transform as the ability of the system to discriminate two different phase velocities. The resolution can be estimated considering an ideal case of a spatial plane wave of phase velocity c n , denoted e pw (f, c n ), with a constant amplitude along the array. In this case, the Norm function is given by the scalar product

( ) ( ) 2 , , pw pw n f c f c e e equal to ( ) 2 R
, sinc

n n c c Norm f c f L c c π ⎛ ⎞ - = ⎜ ⎟ ⎝ ⎠ , (10)
with L R the length of the reception array. Thus, the resolution is given by the size of the main lobe of the sinc function, i.e. fΔcL R / c 2 ≈ 1 , where Δc is the difference between c n and c.

With λ the wavelength satisfying λ = c / f, the resolution condition writes

R c c L λ Δ ≥ . ( 11
)
Previous equation means that two velocities are resolved when their difference Δc/c is larger than the ratio λ/L R . If the amplitude of a guided mode decreases along the array, due to attenuation 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. Thus, the second threshold value t 2 has to be adapted to each experiment.

Furthermore the evaluation domain can be estimated comparing the wavelength λ with two typical lengths: the size of the array L R and the array pitch, equal to L R /N R . First, the Shannon criterion is verified when the half wavelength λ/2 is larger than the array pitch.

Secondly, the half wavelength should also be smaller than the array size. Thus the range (Fig. 4).

C. Experimental results: Singular values and phase velocities

Figure 3 shows the singular values σ n as function of the frequency-thickness fe product obtained with the two probes, 1 MHz (a) and 2 MHz (b),for a 2 mm thick copper plate. In both cases, the first singular value σ 1 is predominant at low frequency, for fe < 1 MHz.mm. It is related to the strong normal displacement associated with the first anti-symmetric A 0 mode. [START_REF] Núñez | Transfer and Green functions based on modal analysis for Lamb waves generation[END_REF] The large peaks correspond to thickness resonance of modes, associated with the cutoff frequencies, as for example the A 1 mode (fe around 1.6 MHz.mm) and A 2 mode (fe around 5.5 MHz.mm). The thin peaks corresponds to the resonances associated with the zero group velocity (ZGV) modes, [START_REF] Prada | Local vibration of an elastic plate and zero-group velocity Lamb modes[END_REF] for the S 2 mode (fe = 2.10 MHz.mm) and S 6 mode (fe = 6.85 MHz.mm). These modes are listed on the upper scale of Fig. 4 showing the corresponding guided mode phase velocities.

The number of significant singular values ranges from 1 to 5 depending on frequency.

That number corresponds to the number m of significant singular vectors used in Eq. ( 8). It depends on the signal to noise threshold t 1 , shown with a curved line. The threshold t 1 has been heuristically fixed for the two probes by searching the best experimental separation between the signal and noise subspaces. 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 The two limits of the evaluation domain described in paragraph III.B are marked. The lowest limit corresponds to Δc/c ≈ 2/N R = 1/7 ≈ 0.13. The highest limit corresponds to Δc/c ≈ 2. Some resolution values (1/4, 1/2, 1) are marked with dashed lines for reference. Following the resolution condition [Eq. (11)], two velocities are resolved when the ratio λ/L R is less than the velocity difference Δc/c. In example (a), A 1 and S 2 modes are non-resolved for fe about 2.2

MHz.mm, with Δc/c ≈ 0.3. In example (b), A 3 and S 1 modes are also non resolved for fe about 4.5 MHz.mm with Δc/c ≈ 0.2. In these two cases the ratio λ/L R ranging from 0.25 to 0.5 is larger than the difference Δc/c. In other cases, the resolution condition is verified and the experimental phase velocities are resolved. The values of the Lamb phase velocities depend on three physical parameters: the thickness e and the transverse and longitudinal velocities, denoted c T and c L . [START_REF] Royer | Elastics waves in solids 1 : Free and guided propagation[END_REF] It is possible to evaluate these parameters by searching the best fit between theoretical and experimental phase velocities, denoted c n th and c n exp , over the whole frequency bandwidth. [START_REF] Wu | Inverse determinations of thickness and elastic properties of a bonding layer using laser-generated surface waves[END_REF][START_REF] Gao | Laser ultrasonic study of Lamb waves: determination of the thickness and velocities of a thin plate[END_REF][37] The agreement is obtained here by minimizing the fitting error defined by the normalized sum of all individual fitting errors for each mode at each frequency The two dimensional spatio-temporal Fourier transform is one of the usual methods used to determine the guided mode phase velocities. [START_REF] Alleyne | A two-dimensional Fourier transform method for the measurement of propagating multimode signals[END_REF] The scalar product

( ) ( ) ( ) ( ) mode max min exp th L T L T exp th 1 L T , , , 1 , , 100 , , , f N n

Conclusion

An original method for evaluating without a priori the guided mode phase velocities, using a probe consisting of two separated emission and reception arrays in contact, is exposed in this paper. The phase velocities are obtained using the received signal subspace defined by the singular vectors of the experimental response matrix. The two main advantages of the method are presented: the signal-to-noise separation and the improvement of the ability to evaluate phase velocity compared to the spatio-temporal Fourier transform. To this end, two specific thresholds are described. Experimental results are in good agreement with the theoretical Lamb modes. These results hold promises in terms of evaluation of the wave guide structural and material properties using limited aperture array, andin terms of bone strength assessment. Further applications will concern in particular the evaluation of elastic properties of cortical bone. Attenuation non-isotropic medium, and variable thickness will be taken into account.

Fig. 1 .

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

  the evaluation domain, where the guided mode phase velocities c n exp can be evaluated. Figure2shows this domain in the frequency-phase velocity (f, c) plane. Thus, each linear function passing through the origin satisfies c = λ f.

Fig. 2 .

 2 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/L R shown as linear function. The boundaries are shown on the experimental results

Fig. 3 .

 3 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 t 1 is shown with a curved line. The singular values above the threshold define the signal subspace.

Figure 4

 4 Figure 4 shows the experimental guided mode phase velocities c n exp corresponding to

Fig. 4 .

 4 Fig. 4. (color online) Phase velocities for 2 mm thick copper plate, evaluated with probes of central frequency f 0 equal to 1 MHz (a) and 2 MHz (b). Theoretical Lamb values are shown in continuous (symmetric modes S n ) and dashed lines (anti-symmetric modes A n ). 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/L R 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.

  with N exp the number of experimental phase velocities, and n the mode label, i.e. A n or S n . The fitting errors are shown in Fig.5for the two probes as a function of c L and c T at a fixed thickness e equal to 2 mm. The fitting error values range from 5 % to 30 %. Minima of the fitting errors yielded values of c T = 2250 m.s -1 and c L = 4700 m.s -1 for both probes. These values are consistent with the usual values for copper.

Fig. 5 .Figure 6

 56 Fig. 5. Fitting error (%) given by Eq. (12) as a function of c L and c T for a fixed thickness e, for a 2 mm thick copper plate, evaluated with probes of central frequency f 0 equal to 1 MHz (a) and 2 MHz (b), respectively.

  interpreted as the spatial Fourier transform of the reception singular vector U n (x R ), the spatial vector k being equal to 2π f/c. The normalized spatio-temporal Fourier transform of the received signals on the array after the first emission, i.e. by a dashed line. In example (a), only one of the three present modes, S 0 is clearly resolved by the spatial Fourier transform. In example (b), the three modes are resolved by the spatial Fourier transform, but the maxima ranges from 0.2 to 1. As a comparison, the maxima of the Norm function are larger than 0.98 in this case. Thus the method proposed here appears to be more robust: the experimental guided modes phases velocities are better evaluated in the whole frequency domain as shown in Fig.4.

Fig. 6 .

 6 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 the first scalar products

Fig. 1 .

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

Fig. 2 .

 2 Fig. 2. The evaluation domain is shown outside the dashed domain. The resolution condition

Fig. 3 .

 3 Fig. 3. (color online) Normalized singular values in dB scale versus frequency-thickness fe

Fig. 4 . 29 Fig. 5 .

 4295 Fig. 4. (color online) Phase velocities for 2 mm thick copper plate, evaluated with probes of

Fig. 6 .

 6 Fig. 6. (color online) Evaluation of the guided modes phase velocities at a fixed frequency: (a)

J. Vishnuvardhan, C. V. Krishnamurthy, and K. Balasubramaniama, "Blind inversion method using Lamb waves for the complete elastic property characterization of anisotropic plates," J. Acoust. Soc. Am. Volume 125,761-771 (2009).

Acknowledgments

This work has been funded by the ANR TecSan. Authors would like to thank Dr. Claire Prada for the fruitful and helpful discussions.