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P-τ TRANSFORMATION AS THE EFFICIENT TOOL FOR DETERMINATION OF THE VELOCITY DISPERSION CHARACTERISTICS IN COMPLEX STRUCTURES

B. Piwakowski^1,M. Goueygou¹, A. Fnine^2,F. Buyle-Bodin²;

¹Institute of Electronics, Microelectronics and Nanotechnology DOAE UMR CNRS 8520;

²Laboratory of Mechanics in Lille, URA CNRS 144, France.

Abstract: Last years show the increasing interest in using Spectral Analysis of Surface Waves (SASW) for the NDE of concrete structures. The dispersion characteristics are there commonly determined using the classic approach based on the Phase Difference measurements (PD) method. Meanwhile, this approach is accurate only for simplest cases. When the investigated signal include different components (modes) if different velocities and if these modes overlap in time domain, the use of the PD may lead to serious errors.

In preliminary studies the authors emphasized on the interest of using the surface waves for determination of the near surface deterioration of concrete. This deterioration may be equivalent to the two-layered structure which involves the modal propagation. In order to determine the dispersion characteristic of each mode, the so-called (P-τ) transformation is proposed. The developed processing code is validated using the simulated input signals with known dispersion law. Then the accuracy of the (P-τ) is studied as the function of the acquisition system parameters like receiver spacing, receiver number, aperture length, sampling rate and signal frequency. The easy to use criteria are formulated.

Introduction: The well known free-space Green function for the elementary case illustrated in Fig.1.a (Kinsler et Frey 1982) yields:

(1.a)

The function G represents the signal observed at the distance R and radiated in unlossy free-space from a point source S, placed in M and excited by harmonic signal

t j

R M G

(

-

) , , ,

π ω

4

e t

=

e R

t j

ω

jkR

e^ω ;

k=ω/V is the wave number and V denotes propagation velocity. For pulsed excitation δ(t) above solution becomes (Harris,1981):

(1.b)

where symbol ℑ indicates the Fourrier transformation. Consequently, signal x(R,t) obtained for the excitation s(t) can be found by means of the convolution :

(2)

As illustrated in Fig.1.b, velocity V can be determined in time domain, measuring delay ∆ t of

signals x(R₁,t) and x(R₂,t). Such measurement yields the group velocity V_g. velocity which

characterizes the total signal. From Eq.(2) the delay can be found as

g

g

g g v

R M G t R M g π

( -

) , ,

ℑ =

-

1 )

( [

)] , ,

δ ω

=

(

V R t

/

4

R

t R g t s t R x
-

) , (
V

R t x R

=

) (

) , ( *

=

1

(

) /

π
4

= - - = ∆ 2

R v

R v

R t ∆

1 and

finally the group velocity can be found as :

t

R V_g

∆ ∆ = (3)

a

R

O

S

M

∆ t g

∆ t e

b

x 1 ( R 1 , t )

x 2 ( R 2 , t )

t

S

Fig.1. (a) Geometry for Eq.(1); (b basic setup for the measurement of signal velocity

In practice, for pulsed signals, the delay ∆t is found either taking first arrivals instants of signals (∆t_e; easy to detect), or taking the delay ∆t_e between their most significant amplitudes (Fig.1.b). In the latter case V_g is in fact supposed to be equal to so called energy velocity V_e which is the velocity of the most energetic signal component.

The signal x can be analyszd in the frequency domain as

) , (

) ( ) ( ω ω ω j

R G j S j X = where:

R j X

( ω

ω

) ,

ℑ = and )]

( [ )

)] , ( [

R X R t x

=

) , (

^ω
exp

-

j

φ

) (

j S ℑ

( t s

=
ω
(4.)

Using the Fourrier transform property

ωτ

- ℑ the phase φ term may be found

( ⇔

s )

ω τ

)

e j S

(

j

as:

= ω ω φ )

, (

R R =

(5)

kR V

Finally velocity V can be found using the setup shown in Fig.1.b, by measuring the phase difference ∆φ =φ₁ - φ₁ of signals x(R₁,t) and x(R₂,t):

V
_ph

) (
ω

φ ω φ φ

2 (6)

The velocity found using this way known as phase velocity V_ph(ω) and the method based on relations (5) and (6) will be further referred as phase difference method (PD). For the no dispersive medium the group and phase velocity are equal i.e V_g= V_ph. In the contrary, when the dispersion of the medium the group and phase velocity are equal i.e V_g= V_ph. In the contrary, when velocity dispersion occurs, the phase velocity is not constant and becomes a function of frequency. Then Eq. (5) assumes more general form:

ω ω

=
R

(

R R

-

1

)

∆ =

-

∆

) (

2

1

ω ω φ

) , (
ω

R
=

R

,

(7) and function V_ph(ω) represents the velocity dispersion characteristic. Notice, that if Eq.(6) is used in order to determine V_ph(ω), then, the phase periodicy yields the infinity of solutions:

V

) (

ph

)

∆
2
(8)

In order to find the unique solution, modulo 2π term should be removed. Then V_ph is found as:

) , (
ω

φ π ω φ
o

n R
∆

+ =

(

ph
R

=
1 2 (9)

-

V
∆

(

) (
ω

φ

ω ω

R

)

) (

o

The PD method is basic and quite popular in NDE field (see for example Cho, 2003; Mathews, 1996; Hassaim et al. 2001; Goueygou, 2003)). Its procedure is illustrated in Fig. 2: the input pulses x₁ and x₂. were modeled numerically, for the dispersive medium characterized by the velocity dispersion V_ph(ω) taken as :

¹
() ()
  

ω π

α ω ω
ln

=

1

-

^Vph V

2 0

  

ω

(10)

o

0

0

where V_o indicates the highest (i.e. group) velocity [above relation characterize the medium with the absorption increasing proportionally to the frequency α=α_of. (Azini,1968)]. Fig.2.a shows two signals x₁ and x₂ computed for a_o==25dB/mMHz, V_o=1000 m/s and s(t)=δ(t). The computations were performed using the DR method (Piwakowski and Sbai, 1999 ; Piwakowski and Lingvall, 2004). Fig.2.b shows phases φ₁ et φ₂ which display, as it could be expected from Eq.(8), the 2π periodicity. Section (c) shows the V_ph(ω) obtained which agrees perfectly with the relation (10) confirming the accuracy of the method.

In spite of advantages its advantages, however, the PD approach also displays several disadvantages:

1. gives uncertain results when signal to noise ratio is poor, 2. the removing of the phase periodicy may lead to certain errors, 3. PD method is not adapted to the more complicated situations occurring when input signals include the different components (modes) which propagate with different velocities. For example this occurs when the structure is layered (modes of Lowe and pseudo-Rayleigh waves). If modes overlap in time their separation in time domain is impossible and use of the PD method may lead to serious errors.

The problem mentioned in (3) is illustrated in Fig 3 on example of the real data. Fig.3.a shows the five input pulses x₁...x₅ recorded at distances R₁...R₅, windowed which Gaussian window in order remove the noise and the not desired signal parts. The input pulses include the different modes displaying the different velocities which overlap in time (i.e. not seen directly). Finally each pair of signals yields different dispersion characteristics (section (b)) and the averaging of the obtained characteristics obviously does not improves the accuracy and leads to the erroneous result without any physical meaning.

2 0

a

p ro g ra m m e : d is p e rs io n 1 d a ta :c :\d re a m 9 t\a 2 5 _ 1 0 .re p

0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1 .4 1 .6 1 .8 2

Fig.2. Evaluation of the dispersion characteristic using PD method ; (a) Two input signals x₁ (thick line) and x₂ (dotted line) computed for absorbing medium α_o=25dB/mMHz (b) Their phases φ₁ et φ₂ ; (c)Dispersion characteristic obtained, observed in interval 0.3-2 MHz

p

0

-2 0

m ic ro s e c

5 1 0 1 5 2 0

radian

)

^{8 5 0}0 2 0 4 0 6 0 8 1 1 2 1 4 1 6 1 8 2

5

b

phase

(

0

-5

c

1 0 0 0

M H z

9 5 0

9 0 0

a

b

c

Fig.3. Results obtained using FD approach (a) Input data, (b)Dispersion characteristics obtained from each pair of input data and the mean characteristic (thick line). (c) Spectres of input data indicating the zone of the validity of results (sufficient S/N ratio)

P-τ transformation: In order to overcome the above mentioned errors the P-τ transformation, known also as the Slant Stack (SL) transformation, used in geophysics, especially in MASW techniques (multichannel analysis of surface waves). (Ymaz, 1987, Leparoux.2002). The input N signals recorded in positions R₁ ...R_n are processed as the two dimensional signal x(t,R). The SL procedure consists in introduction of the linear delay τ=R/V=pR:

τ (11

where the parameter p indicates the slowness:

(12)

The operation is performed in the interval of interest (p_min - p_max). Summing all y in R domain yields the P-τ gather which represents the input data in the slowness- arrival time domain:

) , , ( R pR t x p R y -

=

(

) ,

p 1

=

V

p P

t τ

) , , ( ) , ( p

R x

τ
∑

=

(13)

Rn

R

1

The time domain Fourrier transformation of P_τ and relation (12) convert the input data into (V,ω) domain:

p V

V SL /

) , ( 1

ℑ = τ ω τ (13)

=

)] , ( [

p P

Finally the researched dispersion characteristic V_ph(ω) :is obtained as

V V ph ω

[ max ) ( V

ω =

ST

)] , (

(14)

Figure 4 shows an example of the use of th SL transformation. The six input pulses x₁...x₆ computed by the same mean as those used in Fig.2, are shown inFig.4.a. P_τ plot detects the arrival instant of the input data set (τ_a=30 ms) and shows the interval of detected velocities. The SL plot shows the velocity interval and the frequency bandwidth of input data. The dispersion characteristic found using Eq.(14) perfectly agrees with that defined by Eq.(10).

a b

τ_a

c d

τ_a

Fig.4. Illustration of the SL procedure (a) Input synthetic data, (b) SL(ω,V) plot; thick line marks the obtained dispersion characteristic V_ph(ω) ; (c) P_τ(τ,V) plot showing the input data in velocity -time domain ; (d) V_ph(ω) obtained in from section (b) compared with Eq.14.

Fig.5. Illustration for the SL method (left) input real measured signals, (right) The SL(ω,V) plot indicates the presence of two modes; thick lines indicate their dispersion characteristics V_ph(ω)

Figure 5 illustrates the SL transformation applied to the real measured signal which contains two modes differing in velocity and frequency. Two modes are clearly separated and their separate dispersion characteristics are obtained. Comparing with the FD method the SL transformation offers the following advantages: - V_ph(ω) is obtained from total set of data , and not from two signals only, thus the signal /noise ratio is improved and the averaging of individual characteristics is not necessary. - different modes can be visualized, identified , and analyzed both in (V,ω) and in (V,t) domains, - the operations (17) and (18) are reversible thus allowing a mode extraction and return into time domain - the group and energy velocity can be determined in the same time, directly from the SL plot.

θ

d

λ

x x x x x ... ... . x x

N-1 N

L = (N -1 )d

1 2

Fig.6. Linear array geometry equivalent to SL operation

Accuracy: The problem of accuracy of SL transformation is generally ignored by its users. In the same time several authors consider the approach SL as to be a aliasing free, and this statement seems be physically doubtful.

The included examples show that the accuracy of the determination of the measured velocity is always limited and is directly related with the width of cross section of SL in velocity domain. In this section the accuracy of the SL method will be formally evaluated. In order to study this we introduce the velocity amplitude function A(V,V_s) defined as the cross section of SL plot obtained for the input signal having constant velocity V_s and constant frequency ω=ω_s.

= ω

ω (15)

The SL procedure can modeled as the N element linear array of length L, receiving the planewave

)

(

V SL V V A =

) , ( s input s

s V

, (

=

,

V

)

-

^xtks e

) , ( s s Rk t

j

⁼ω _;ks=_ωs_/Vs , of wavelength _λs_=Vs_/fs , arriving from direction θ=π/2, (Fig.6). Modeling the operations expressed in Eqs.(11),(12) and (13) using the linear array model the array output A(V,V_s) yields: (16)

ω =

sin(

s

dB N

2

)

V V A

) , (

s ω

dB N

sin(

s

)

2

where )

/ ( V

V

/

B s 1

⁼1 - and d is the element spacing.

As it can expected from the array theory the A(V) is the function of two main array parameters: the spatial sampling density d/λ and the array length with regard to the wavelength L/λ/. The figures 7 and 8 show the A(V) plots as the function of the above two parameters. The analysis of these results leads to the following conclusions:

* if the spatial sampling condition criterion d/λ_s < 0.5 is not fulfilled the SL transformation generates the accurate result which is accompanied by false results (ghost images) and their number increases when d//λ_s increases (Fig.7.a). If undersampling is important the accurate result may even interfere with the false results (Fig.7.d). * Taking into account the evident relation λ=V/f_s, and that the analysis is performed as a function of V, the sampling criterion varies during analysis. Finally the results can be corrupted for d<V/2f_s ,(i.e. under the line seen in Fig.7.b,c,d) or in velocity interval defined as V < d/2f_s (see lines shown in Fig.7.e and Fig.8.e and the ghost images present under these lines). * Figure 8 provides the study of ∆V_3dB as the function of array length/wavelength ratio. As it could be expected, accuracy increases when Nd/λ_s increases. By analogy with the filter theory, we define the 3dB width ∆V_3dB of the main lobe of A(V) observed in vicinity of V_s as the accuracy measure (∆V_3dB is illustrated in Fig.8.b). The solution (16) enables to approximate accuracy as :

a

)

/

,V

1

d

λ

3

A (

0 .5

²d / λ

2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0

1

0

V_s

V ( m /s )

b c

d e

d/
λ
_s
=0.5

0 .8

0 .8

d/
λ
_s
=2

A (V

A (V

⁾0 .6

⁾0 .6

0 .4

0 .4

0 .2

0 .2

d=V
_s
/2f
_s

⁰200 0 40 00 60 00 8 000 1 000 0

V
_s

⁰200 0 400 0 60 0 0 800 0 100 00

V
_s

c (m /s )

c (m /s)

0.8

d/
λ
_s
=4

d/
λ
_s
=1

0.6

A(V)

V
_s

V=2df

0.4

0.2

⁰2000 4000 60 00 8000 10000

V
_s

c (m /s)

Fig.7. Accuracy as the function of the sampling factor d/λ_s. (a) A(1000<V<1000,0.5<d/λ<4) V_s=3000 m/s ; (b,c,d) A(V) from section (a) for three factors d/λ_s Dashed line shows the sampling limit d<V/2f_s; (e) SL(V,f) for d/λ_s =1. Sampling limit assumes here the form of the line V=2df

≈ ∆

V

3

Nd

V s dB 0.81 (17)

The above formula provides the precision close to 2% when ∆V_3dB / V_s is smaller than 5%.The plot of the ratio ∆V_3dB / V_s is shown in Fig.8.d. It may be concluded that the accuracy better than 10% requires the array length L/λ better than to 10.

Conclusions: The performances of the SL transform has been evaluated and accuracy of the velocity estimation was studied and expressed quantitatively, as a function of the array sampling density and the array length in regard to wavelength.

It is shown the SL transformation is not aliasing free, as it is reported in certain references. In fact even if the sampling criterion is not fulfilled, the result can be correct, but it will be accompanied by the false detections of the false, not existing velocities. Thus effect is difficult to avoid because the sampling criterion varies with velocity which, by principle, varies during the analysis. Finally during the use of the SL method, the sampling criterion should controlled by the operator, for example by drawing the sampling limit curve V=2df like it was done in Figs.7 et 8.

The accuracy better then 10% requires the array length/wavelength ratio to be of order of ten .

-

1

λ

a

∆
V
_3dB

V
_s

b c

d e

∆
V
_3dB

Nd/
λ
_s
=5

Nd/
λ
_s
=40

V
_s

V
_s

100

80

∆
V
_3db
/V
_s

V
_s
=2df

2.5 5 10 25

∆ c /c %

60

V
_s

40

20

Nd/λ_s

⁰2 5 10 15 20

Nd/λ_s

Nd/λ

Fig.8 Accuracy as the function of the array length Nd/λ_s. (a) A(1000<V<1000,1.5<Nd/λ<15) V_s=3000 m/s ; (b,c,) A(V) plots from section (a) for two factors Nd/λ_s Accuracy ∆V_3dB ls illustrated in (b). (d) Relative accuracy ∆V_3dB / V_s as a function of Nd/λ (Eq.17); (e) SL(V,f) for d/λ_s =1. Sampling limit assumes here the line V>2df

Acknowledgments: This research is supported by the program "Urban and Civil Engineering Network" (RGCU) of the French Ministry of Education and Research.

References : - Azimi, Sh "Impulse transient characteristic of media with linear and quadratic absorption laws," Izvestija, Physics of the Solid Earth, pp. 88-93, 1968. - Cho,Y,S, Lin,F.,B. (2001) " Spectral analysis of surface wave response of multi-layer thin cement mortar slab structure with finite thickness" NDT&E International 34, pp. 115-122. - Harris, G. R. (1981) « Review of transient field theory for a baffled planar piston » J. Acoust. Soc. Am. 70(1), pp 10-20. - Hassaim,M J. Rhazi, G.Ballivy et K. Khayat, " Evaluation de l'état du béton par technique d'analyse spectrale des ondes de Rayleigh. Can.J.Civ.Eng. 28, 2001, pp 1018-1028. - Kinsler,L,E, Frey,A,R . " Fundamentals of Acoustics". John Willey & Sons, 1982 - Lingvall, F., Piwakowski., B. "DREAM free software": http://www.signal.uu.se/Toolbox/dream - Ould Naffa, S. "Evaluation de la dégradation du béton par ondes ultrasonores " PhD disertation. University of Valenciennes, France 2004. - Papoulis, A "Systems and Transforms with Application in Optics" Krieger Publishing Co., Inc. 1981 - Piwakowski, B. and Sbaï, K. (1998) « A new approach to calculate the field radiated from arbitrarily structured transducer arrays » IEEE Trans. Ultrason., Ferroelect., Freq. Contr 46(2) 1999, pp 422-439. - Yilmaz O, (1987) Seismic Signal Processing, Society of Exploration Geophysicists,