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Identification of Shear Wave Parameters of Viscoelastic Solids byLaboratory Measurements of Stoneley-Scholte Waves

Nathalie Favretto-Anres Jean-Pierre SessaregoCentre National de ]a Recherche Scientifique Laboratoire de Mecanique et d Acoustique 3] chemin Joseph Aiguier 13402 Marseille Cedex20 France

SummaryThis paper deals with the problem of viscoelastic solid characterization by acoustical means and in particular with therecovery of the shear wave parameters It has been previously shown in the Underwater Acoustics field that the shearwave parameters of the sea floor could be recovered by using inverse techniques applied to propagation characteristics ofinterface waves such as Stoneley-Seholte waves which can propagate in watersedi ment configurations The goal of thestudy presented in the paper is then to test models commonly used for seabed identification on media whose propertiesarc well-controlled by laboratory tank experiments contrary to in situ bottoms It is shown that the viscoelastic mediumparameters can also be identified from the characteristics of interface waves generated experimentally in laboratoryon very attenuating synthetic materials The paper presents results about the estimated shear wave parameters obtainedfrom both numerical and experimental data by applying Brents method on the characteristics of the interface wavesThe observation and the discussion of differences between theoretical and experimental results are the goa] of the paperThe study presented here validates the forward model previously developed and it can be considered as a first steptowards the direction of acoustic classification of sea bottomsPACS no 4330Mo 4335Mr 4335Pt

1 Introduction

The physical properties of sea floor sediments is of greatinterest for marine geologists people working in the under-water acoustics domain as well as people involved in oilexploitation or in industrial fisheries The applications coverthe field of marine environment geotechnical research civilengineering and underwater acoustics Usual techniques (di-rect in situ techniques acoustical probing ) lead to thedetermination of the compressional wave parameters and thedensities of the layers [I 2] and sometimes to the determina-tion of the shear wave velocity of the sediments [3] Over thepast two decades an increasing effort has been put into thisfield Recent developments in forward and inverse modelingindicate that the acoustic and seismic waves in sea floor canbe used for remote sensing of the geotechnical properties ofsediments In particular the propagation of interface wavesobserved in such an environment is strongly dependent onthe physical characteristics of sea bottom Since the velocityof both Love waves and Stoneley-Scholte waves called alsoScholte waves is strongly linked to the shear wave velocityof the sedimentary bottom one can easily obtain informationabout the shear wave velocity of sediments by applying in-verse techniques to the dispersion curves of interface waves[4567] Few works are concerned with the determinationof the shear wave attenuation in the sedimentary layer [81

Even if the different models commonly used for identifi-cation of sediment properties seem to be reliable a questionmay be however raised how to be sure of the accuracy of the

Received 3] March 1998accepted 26 April 1999

present adress Laboratoire dImagerie Geophysique IPRAUniversite de Pau et des Pays de ] Adour Avenue de ]Universite64000 Pau France

estimated parameters of non-controlled media The aim ofthe paper is then to give an answer to the question by consid-ering the possibility of recovering the well-controlled solidparameters with the propagation characteristics of Stoneley-Scholte waves in waterabsorbing solid configurations Thesolid medium was represented by attenuating synthetic resinsand the interface wave properties were obtained from exper-iments performed in laboratory

The first author has previously conducted theoretical andexperimental studies of the Stoneley- Scholte wave propa-gation in waterviscoelastic solid configurations [9 10] Theoriginality ofthe studies was to release the assumption of nonabsorbing media and to suppose the solid bOltom to be highlyviscoelastic The medium was represented by a Kelvin- Voigtsolid which is generally described by a parallel connectionof stiffness and viscosity elements The viscosity coefficientsintroduced in the constitutive law were then connected to theplane wave attenuations in the solid which can easily be deter-mined by acoustical measurements Analysis of the interfacewave propagation was developed by using the approach of in-homogeneous plane waves which is really suitable for such astudy It enables one to locally describe the interface wave asa linear combination of inhomogeneous plane waves whichsatisfy boundary conditions at the interface in each mediumIt was shown that the wave is dispersive that the energy isgenerally concentrated in the fluid medium and that it de-creases with increasing distance from the interface [9] Theoriginality of the study was to compute the velocity of theinterface wave and the damping of its energy along its prop-agation path at the waterviscoelastic solid interface Theseproperties were also verified experimentally in laboratorywith absorbing synthetic resins [10] which demonstrates thevalidity of the theoretical model previously developed Someof the propagation characteristics of Stoneley-Scholte waveare reviewed in Appendix at the end of the paper

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Figure I Effect of the changes in the shear wave velocity of theviscoelastic medium on the dispersion curves of the phase velocityof the Stoneley-Scholte wave (a) At the waterPVC interface --dispersion curves corresponding to the exact values of the shear wavevelocity of PVC material dispersion curves corresponding to adecrease of the shear wave velocity by 15 0000 dispersion curvescorresponding to an increase of the shear wave velocity by 15 (b)At the waterfB2900 interface dispersion curves correspondingto the exact values of the shear wave velocity of B2900 material0000 dispersion curves corresponding to a decrease of the shearwave velocity by 15 dispersion curves corresponding to anincrease of the shear wave velocity by 15

Figure 2 Effect of the changes in the shear wave attenuation of theviscoelastic medium on the dispersion curves of the phase veloc-ity of the Stoneley-Scholte wave (a) At the waterPVC interface dispersion curves corresponding to the exact values of the shearwave attenuation of PVC material - - dispersion curves corre-sponding to a decrease of the shear wave attenuation by 15 - - -dispersion curves corresponding to an increase of the shear waveattenuation by 15 (b) At the waterfB2900 interface -- dis-persion curves corresponding to the exact values of the shear waveattenuation of B2900 material dispersion curves correspondingto a decrease of the shear wave attenuation by 15 0000 dispersioncurves corresponding to an increase of the shear wave attenuationby 15

Before solving the inverse problem that is the determina-tion of the shear wave parameters of the viscoelastic solidsfrom the propagation characteristics of the interface wave aparametric study of the influence of the physical propertiesof solids on these characteristics was necessary Section 2presents the results of this parametric study These findingswere then used to develop an inversion strategy

Finally Section 3 is devoted to the solution of the inverseproblem Inversion of interface wave data was carried outusing Brents algorithm The inverse scheme was able torecover the shear wave velocity and attenuation of the bot-tom assuming an a priori knowledge of the environment Italso assumed horizontally layered media (plane and paral-lel interfaces) and homogeneous layers Several test resultsare presented in this section The originality of the study

is that the inverse algorithm applied to either dispersion orenergy curves was tested against both noisy synthetic dataand data obtainedfrom experiments performed in laboratoryThe observation and the discussion of differences betweentheoretical and experimental results in the latter case maythus be a good criterion for the validity and applicability ofthe scheme performed under in situ conditions

2 Introduction to inverse problem

It is evident from previous works on forward modeling[9 10 ] 1] that the parameters of the shear waves in vis-coelastic medium conditione the propagation characteristicsof the Stoneley- Scholte wave at the ideal fluidviscoelastic

ACUSTlCAmiddot acta acusticaVol 85 (1999) Favretto-Anres Sessarego Identification of viscoelastic solid parameters 507

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Figure 3 Effect of the changes in the shear wave attenuation of the viscoelastic medium on the energy curves of the Stoneley-Scholte wave atthe waterPVC interface (a) Frequency 30 kHz wavelength = 3 cm (b) frequency 80kHz wavelength = 11 cm (c) frequency 130 kHzwavelength = 07 cm (d) frequency 200kHz wavelength = O4cm energy curves corresponding to the exact values of the shear waveattenuation of PVC material 0000 dispersion curves corresponding to a decrease of the shear wave attenuation by 15 dispersioncurves corresponding to an increase of the shear wave attenuation by 15

solid interface In particular if the wave attenuations in thesolid bottom are taken into account the interface wave isdispersive and is attenuated during its propagation at the in-terface (see Appendix) The goal of this section is to report asensitivity study based on the forward model (see Appendix)and to determine which parameter can be recovered by inver-sion methods In fact what is important to know here is thesensitivity of the propagation characteristics of the interfacewave to small changes in the shear wave velocity or in theattenuation of the viscoelastic medium

21 Effect of the changes in the shear wave parameters onthe dispersion curves

Figures I and 2 illustrate the dispersion curve of thephase velocity of the Stoneley-Scholte wave in differentfluidviscoelastic solid configurations and in particular atthe waterPVC interface and at the waterB2900 interface(see Appendix)

Figure I corresponds to an increase and a decrease of theshear wave velocity by an amount of 15 in comparisonwith the exact values In both cases the shear wave velocity

is an important parameter for the dispersion curves a smallvariation of the velocity results in a variation of about 10of the dispersion curve of the Stoneley-Scholte wave

Figure 2 represents variations of the dispersion curves ofthc Stoneley-Scholte wave when the nominal shear wave at-tenuation of the two solids is changed by factors of 15The behavior of the dispersion curves shows that the phasevelocity of the interface wave is not sensitive to changes inthe shear wave attenuation Therefore shear wave attenuationcan be considered as a passive parameter with respect to thedispersion of the interface wave which has been observed inmany other previous works As the interface wave is sensi-tive to the shear wave velocity of the viscoelastic solid thisparameter will be estimated accurately from the dispersioncurve analysis contrary to the shear wave attenuation

22 Effect of the changes in the shear wave attenuation onthe energy curves

The curves illustrated by Figures 3 and 4 represent the de-crease of the energy of the Stoneley- Scholte wave duringits propagation in different fluidviscoelastic solid configura-

ACUSTlCAmiddot acta acustica508 Favretto-Anres Sessarego Identification of viscoelastic solid parameters Vol R5 11999)

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40 40

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Figure 4 Effect of the changes in the shear wave attenuation of the viscoelastic medium on the energy curves of the Stoneley-Scholte waveat the waterIB2900 interface (a) Frequency 30kHz wavelength = 43cm (b) frequency 80kHz wavelength) = 16cm (c) frequency130 kHz wavelength = 07 em (d) frequency 200 kHz wavelength = 04 em energy curves corresponding to the exact values of theshear wave attenuation of B2900 material 0000 dispersion curves corresponding to a decrease of the shear wave attenuation by 15 dispersion curves corresponding to an increase of the shear wave attenuation by 15

tions particularly at the waterPVC interface and at the wa-terB2900 interface The energy curves of the interface wavefor the frequencies 30 kHz 80 kHz 130 kHz and 200 kHzcorrespond to an increase and a decrease of the shear waveattenuation by an amount of 15 in comparison with theexact values The variations between the values and the exactones are then approximately equal to 4 for each frequencyand for each configuration

The propagation characteristics of the Stoneley-Scholtewave are much less sensitive to the changes in the shearwave attenuation of the viscoelastic medium than to the onesin the shear wave velocity Nevertheless the estimation of theattenuation will be accurate enough for a rough identificationof the solid properties

3 Inverse problem

The problem under consideration is the determination ofthe shear wave parameters of viscoelastic solids fromboth theoretical and experimental propagation character-istics of Stoneley- Scholte waves propagating at ideal

fluidviscoelastic solid interfaces The observation and thediscussion of differences between theoretical and experimen-tal results are the goal of the section

31 Some definitions

It is well-known that the propagation characteristics of inter-face waves are fairly insensitive to density and compressionalparameters of media and that they are function of shear waveparameters of solids only In the previous section the effect ofsmall changes in the shear wave parameters of homogeneousmaterials on the characteristics of the Stoneley-Scholte wavehas been studied From these results it can then be concludedthat it is possible to determine separately the shear wave ve-locity and the shear wave attenuation of the medium fromthe dispersion curves of interface waves and their energycurves Techniques for estimating the shear wave velocityof solids generally consist in measuring the dispersion ofinterface waves at different frequencies and using this in-formation to recover the unknown parameter In a similarway the shear wave altenuation of solids can also be inferredfrom the estimation of the energy loss of interface waves

ACUSTICAmiddot acta acusticaVol 85 (1999) Favretto-Anres Sessarego Identification of viscoelastic solid parameters 509

Table 1Estimation of the shear wave velocity Ct of PVC and B2900 materials from noisy data issued from dispersion curves of the Stoneley-Scholle wave at the watersolid (PVC or 82900) interface for different initial estimates of Ct (In the PVC case the actual Ct is equal to1100 mis and in the 82900 case the actual Ct is equal to 1620 mls)

Initial value of c (ms) 075 Ct 085 Ct 095 c 105 Ct 115 Ct 125 Ct 135 Ct 145 Ct 155 Ct 165 Ct

PVC estimate of Ct (mls) lO98 llO4 lO91 lO95 lO97 lO98 IlO3 lO99 lO96 1]03

estimation errors lt1

82900 estimate of Ct (mls) 1686 1621 1580 1694 1639 1551 1532 ]552 1558 1532

estimation errors 4 lt1 25 46 1 43 54 42 38 54

Figure 5 The schematic diagram of the inverse scheme

(2)

410

lO57

1547

45

estimate of Ct (ms)

estimation errors

estimate of C (ms)

estimation errors

1 N 2

D = N L (YexPk - ~alk)k=]

B2900

PVC

A number of algorithms exist for solving this least squaresminimization problem [13] As there is only one parameterto be identified at a time we chose the Brents method whichis a line search method In fact the method is a combinationof the Golden Section Search and the inverse parabolic in-terpolation [13] Note that in the case of multilayered mediaie when several parameters have to be recovered simultane-ously the Brents method cannot be applied In this case onecan employ a neural network approach [14] for example

Starting from an initial estimate for the shear wave param-eter (velocity or attenuation) equation (2) was iterativelysolved until the distance between the simulated data Yca1k

and the measured data Yexpk came below some specifiedthreshold value or until the difference between two consec-utive updates of the parameter was negligible for all com-ponents of X (see Figure 5) Because of the possibility ofreaching a local minimum of the cost function the inversionhad to be repeated with different initial estimates of the shear

viscoelastic bottom the shear wave velocity of the mediumhas to be identified at first (the shear wave velocity of solidscan be well estimated from dispersion curves even if the at-tenuation is not determined with accuracy) Finally a prioriinformations concerning the solution of the inverse problemsuch as limiting physical values of the parameter had to beintroduced into the algorithm (see Figure 5) The estimationproblem then reduces to the determination of the parameterX which satisfy equation (I) in the least-square sense andwhich then minimizes the cost function D

Table II Estimation of the shear wave velocity Ct of PVC and B2900materials from experimental data issued from dispersion curves ofthe Stoneley-Scholte wave at the watersolid (PVC or B2900) inter-faee for initial estimates of Ct 075 Ct 085 Ct 095 Ct 105 Ct 115Ct 125 C (In the PVC case the actual Ct is equal to 1100 mis andin the B2900 case the actual Ct is equal to 1620 m1s)

(1)

---bull~ Passive -

parameters )---- --_

OS k S N

~Initial estimate-( of the parameter ) X

1-(E~timation ofgtI a new value 1 of X 1

Final estimation -( of the parameter )_ X

where F relates the observed data to the unknown parame-ters F can be the dispersion relation or the function express-ing the decrease of energy of interface waves during theirpropagation The sensitivity analysis of Section 2 clearlyshows that this problem is well-posed Different homogenousmaterials give origin to different dispersion or energy curvesof interface wave The solution is then guaranteed in terms ofexistence and uniqueness As the propagation characteristicsof interface waves are sensitive to shear wave parameters ofsolids only some geoacoustic properties such as density andcompressional parameters of media were then considered aspassive parameters [12] They were assumed to be constantduring inversion To estimate the shear wave attenuation of

during their propagation as it will be shown in the secondpart of the section The relationship between the N measureddata Y k (0 S k S N) (ie phase velocities Ccentk of interfacewave respectively amount of energy lost Ek) function of thequantities W k (ie frequencies fk respectively distances ofpropagation dk) and the unknown parameter X (ie shearwave velocity Ct respectively shear wave attenuation at ofsolids) can be written in the form of a non linear system ofIV equations

510 Favretto-Anres Sessarego Identification of viscoelastic solid parametersACUSTICA acta acustica

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Table III Estimation of the shear wave attenuation at of PVC and B2900 materials from noisy data issued from energy curves of theStoneley-Scholte wave at the watersolid (PVC or B2900) interface for different initial estimates of at (In the PVC case the actual at is equalto 80dBm for 60kHz in the B2900 case the actual at is equal to 7] dBm for 150kHz)

Initial value of at (ms) 075 at 085 crt 095 at 105 at 115 at 125 at

PVC 60kHz estimate of at (dBm) 91 83 83 86 90 90

estimation errors 13 4 4 7 12 13

82900 150 kHz estimate of at (dBm) 79 77 77 67 75 82


Table IVEstimation of the shear wave attenuation at of PVC andB2900 materials from experimental data issued from energy curvesof the Stoneley-Scholte wave at the watersolid (PVC or B2900)interface for different initial estimates of at and for different numbcrof input data for the inverse algorithm (In the PVC case the actualat is equal to 88 dBm for 75 kHz in the B2900 case the actual at isequal to 57 dBm for 70 kHz) Initial value of at (dBm) from 075at to 125 at

PVC 8 88 08 88 0

B2900 3 71 253 65 14

8 86 518 65 14

II 82 44II 65 14

21 9] 6021 65 14

40 50 ]240 65 14

32 Estimation of the shear wave velocity

The inverse procedure for determining the shear wave veloc-ity of viscoelastic solids (PVC and 82900) modeling sed-iments was applied to dispersion curves of the Stoneley-Scholte wave The actual shear wave velocity was equal to1100 ms for PVC material and 1620 mls for 82900

The inverse algorithm was first tested against noisy inputdata These data were obtained by perturbing the theoreticaldata (see Figure A3) in a random way within a percentage

wave parameter to determine if the solution was a true globalminimum liS] The input data for inversion algorithm wereobtained by sampling either the dispersion or energy curvesof interface waves at discrete points in the frequency spaceor in the propagation path space A total of 8 data pointswere generally used in the inversion The inversion was re-peated 20 times for each initial estimate of the shear waveparameter and for each set of input data in order to mini-mize erroneous solutions The final estimate of the unknownparameter was then obtained by an average of the estimatesfor each inversion process The inverse procedure for determining the shear wave atten-

uation of viscoelastic solids (PVC and 82900) was applied tothe energy curves of the Stoneley-Scholte wave It has beenshown in the previous section that the effect of the shearwave attenuation on the energy curves of the interface wavewas the same whatever the frequency Therefore only resultsobtained for a given frequency are presented here The ac-tual shear wave attenuation of PVC was equal to 80 d8m for60 kHz while the attenuation of 82900 was equal to 71 d8mfor 150kHz

The inverse algorithm was tested against noisy input dataThe input data were obtained by modifying the theoretical

of 10 from the exact values The random sequence wasobtained through the use of the function Rand (Matlabreg)which generates uniform distribution of random numbersResults are given in Table I for different initial estimates ofthe shear wave velocity Ct Note that one has to be careful withthe numerical solution of characteristic equation of intcrfacewave by using different initial values of Ct which can bebelow or above the water sound speed In particular as theStoneley-Scholte wave velocity is always below the slowestvelocity of waves in media one has to take a close look to thebranch cuts associated with the square root functions presentin the characteristic equation (A I0) (see Appendix) whichhave not to cross the region of the complex plane where thezeros of equation (AlO) are see ked [9] In Table I it can beseen that whatever the initial estimate for Ct the shear wavevelocity of materials is identified with a great accuracy Notethat the inversion error is always below I in the PVC caseand always below 6 in the 82900 case

The inverse algorithm was also tested against experimen-tal input data obtained from tank experiments [10 II] As nodispersion of the Stoneley-Scholte wave was noted the inputdata for algorithm were constant and equal to 866 ms forthe PVC case and equal to 1260 mls for the 82900 case (seeAppendix) A total of 8 points were used in the inversionSolving the inverse problem then implies to recover the shearwave velocity of viscoelastic media Results are given in Ta-ble II for different initial estimates of the unknown parameterWhatever the initial value of c the shear wave velocities ofPVC and 82900 are determined with accuracy The estima-tion errors are below 5 in the two cases which is in a goodagreement with the inversion results from simulated data

33 Estimation of the shear wave attenuation

Estimationerrors

Estimate of at(dBm)

Number of input data(exp points) (least-square)

ACUSTICAmiddot acta acusticaVol 85 ([ 999) Favretto-Anres Sessarego Identification of viscoelastic solid parameters 511

data (sec Figure A7) in a random way within a percentage of50 from the actual values Results are given in Table III fordifferent initial estimates of the shear wave attenuation atIt can be seen that although the estimation errors are quitegreat (below ]3 for the PVC case and below 16 for theB2900 case) the identification of the shear wave attenuationof materials is accurate enough for a rough identification ofviscoelastic solid properties

The inverse algorithm was also tested against experimen-tal data obtaincd from tank experiments The input data forinversion algorithm were selected from points of the ex-perimental energy curves of the Stoneley-Scholte wave (seeFigure A7) The inversion gives the shear wave attenua-tion of viscoelastic media for a given frequency The actualshear wave attenuation was 88 dBm for 75 kHz for PVC and57 dBm for 70 kHz for B2900 Results are given in Table IVfor different initial estimates for the unknown parameter andfor different numbers of experimental points used as inputdata Whatever the value of the initial estimate the shearwave attenuation of PVC matcrial is identified with a highaccuracy owing to the high quality of the experimental re-sults obtained from tank experiments The estimation errorsare close to 0 On the contrary the shear wave attenuationof B2900 material is not recovered with accuracy whateverthe number of input data the inversion errors are thus closeto 50 This discrepancy comes essentially from the fact thatin the B2900 case each experimental point can be very differ-ent from the theoretical one although experimental data areglobally close to the theoretical curves Has the differencein the shear wave velocity of the two materials (PVC hasS-wave velocity less than water sound speed B2900 higher)anything to do with it Such a conclusion cannot be drawnas long as experiments with another material with S-wavevelocity higher than water sound speed are not performed Inthe B2900 case the lack of correlation between experimentaldata implies bad estimation of the shear wave attenuation ofthe viscoelastic solid The discrepancy can be overcome byincreasing the number of input data of the inversion algo-rithm using I I points (respectively 40 points) in the inver-sion leads to an estimation error close to 44 (respectively12) (see Table IV) But increasing the number of input dataof the inversion algorithm can sometimes imply convergenceproblems of algorithms Therefore the better way to solve theinverse problem when the material properties are not knownis to select points from the least-square line which fits theexperimental data The correlation obtained from measureddata thus implies a quite good estimation of the shear wave at-tenuation of the viscoelastic material In the B2900 case theestimation errors are then close to 14 whatever the initialestimate of at and whatever the number of input data Theseresults are thus in a good agreement with those obtained fromsimulated data (see Table III) In the PVC case the shear waveattenuation of material is perfectly identified owing to thevery good agreement between experimental results obtainedfrom tank experiments and theoretical predictions In partic-ular the least-square line which fits the experimental datais identical to the theoretical energy curve of the Stoneley-Scholte wave The Brents method has been well suited for

our purpose of demonstrating the possibility of recoveringthe shear wave parameters of viscoelastic solids from disper-sion or energy curve data However the estimation generallydepends on the initial estimates of the unknown parametersA better way to solve the inverse problem would be to usethe neural network approach which would be suitable forthe determination of both the shear wave velocity and theshear wave attenuation in solids No initial estimate of theunknown parameter close to the exact one and no a prioriknowledge of the environment are required [14 16] Never-theless the study presented in this section has proved the highinterest in modeling underwater acoustics problems by tankexperiments on well-controlled media in laboratory It can beconsidered as a first step towards acceptable identification ofgeoacoustical properties of viscoelastic media

4 Conclusion

Many other previous works have shown that it is possible todetermine the geoacoustical properties of sediments by re-mote techniques Interface waves such as Stoneley-Scholtewaves observed over specific propagation paths at the wa-tersediment interfaces may contain sufficient information todeduce the properties of solids and in particular the shearwave parameters of sedimentary bottom But how to be sureof the accuracy of the estimated parameters of non-controlledmedia The aim of the paper was then to give an answer tothis question and to show that testing the performance ofinversion methods on synthetic materials which can modelmarine sediments might be a very useful tool and mightthen be applied to the Underwater Acoustics field It couldbe considered as a first step before the final ground thruthperformed at sea under realistic conditions In the study weconsidered the possibility of recovering the parameters ofwell-controlled media with the propagation characteristicsof Stoneley-Scholte waves at waterabsorbing solid inter-faces In particular we used in inversion studies experimentaldata obtained from tank experiments It has been shown thatthe shear wave velocity of solids could be reliably identifiedfrom dispersion curves of interface waves A strategy fordetermining the shear wave attenuation of solids has beendiscussed when the material properties are not known Theestimation of the shear wave attenuation from the energycurves of interface waves expressing the amount of energylost along a propagation path was then accurate But the es-timation quality seems to depend on the solid materials Theconclusion that the difference in the shear wave velocity ofthe two materials (PVC has S-wave velocity less than wa-ter sound speed B2900 higher) has maybe something to dowith it but it cannot however be drawn so long as experimentswith another material with S-wave velocity higher than watersound speed are not performed

Our method has been developed under the assumptionof viscoelastic media In future works and in particular forpeople interested in the study of marine sediments it wouldbe necessary to modify our theory and experiments in orderto consider poroviscoelastic or inhomogeneous viscoelasticmedia

512 Favretto-Anres Sessarego Identification of viscoelastic solid parametersAClJSTlCA aela aeuslea

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Figure A2 Measured attenuations of the plane waves in B2900 mate-rial (a) longitudinal waves (b) shear waves 0 experimental valuebullbull-- the slope obtained by the least-squares fit of the experimentaldata I error ranges

Appendix general form

The subscript vI = L T is used to denote the lamellar andtorsional waves u~j is a complex-valued constant vector andK~f the complex-valued wave vector of the inhomogeneouswaves The real-valued vectors K~ and K~f respectivelyrepresent the propagation and the attenuation of the wavekm are the wave numbcrs of thc associated homogeneousplane waves (longitudinal (m l) and shear (m t) ones)The quantities Vm are defined by the ratio between the vis-cosity coefficients (A and 1I) and the Lame coefficients(1 and M)

The propagation of the Stoneley-Scholte wave at the idealfluidviscoelastic solid interface has been studied in previouspapers [9 10 IIJ Only a synthetic summary is presented inthis section Since the analysis of the propagation of inter-face wave was developed by using the approach of inhomo-geneous plane waves it seems necessary to recall first someproperties of the waves in viscoelastic r 17]media and in fluidmedia [18]

A1 Review of some properties of inhomogeneous planewaves

A 11 In a viscoelastic solid medium

Two inhomogeneous plane waves can propagate into a solidmedium assumed to be viscoelastic homogeneous isotropicand infinite a lamellar one (curluL = 0) and a torsionalone (divuT = 0) The acoustic real-valued displacementsuLand UT associated with these waves are sought in the

with

U~f

KrvI

K~l K~f

U~1exp(iK~l r)

K~ +iK~fkW(l + iwvm)

2lVI + Aper

(AI)

(A2)

(A3)(A4)

(A5)

ACUSTlCAmiddot acta acusticavou (1999) Favretto-Anres Sessarego Identification of viscoelastic solid parameters 513

950

940~~

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gt 920~2910sect~~ 900c0-

890

880

870

0

1440-

20 40 60 80 100Frequency (kHz)

120

(a)

140 160

0025(a)

002

0015

~001

~ 0005

~laquo

05 15 2 25 3 35 45Ti me (second)

x 10

Figure A3 Theoretical dispersion curves of the phase (--) andgroup ( ) velocities of the Stonelcy-Scholte wave (a) For thewaterPVC interface (b) for the waterfB2900 interface

1420

1400-

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-0

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1280

126W

12401~ _o 20 40 60 80 100

Frequency (kHz)120

(b)

140 160

~t-005

05 15 2 25T I me (second$)

(b)

35

I

x 10

Figure AS Signals detected by the receiver (a) at the waterPVCinterface wedge-receiver distance = 6 cm (b) at the waterfB2900interface wedge-receiver distance = 38 cm

Figure A4 Schematic representation of the experimental set up

(A7)

(A8)

2al(W)wkl(w) 2ot(w) cos 8

wkt(w)

8 is the refracted wave angle in the solid material [11]Equation (A4) implies that the vectors representing prop-

agation and damping of the inhomogeneous plane waves arenot perpendicular in this case the waves are called heteroge-neous waves

geneous longitudinal (respectively shear) plane waves r isthe space vector and w is the angular frequency

The quantity VI (respectively Vt) can be expressed in termsof the measured attenuation of the longitudinal homogeneous(respectively transversal inhomogeneous) plane waves inthe solid

Tank

Receiverllll=l mmgt

ater

Pulse GeneratorSOFRAIEL

Tetlowwedge+

Shear wavetramducer

(A6)

p being the density of the solid and C(m~l) = y(2M + A)j p

(respectively c(m~t) = yMj p) the velocity of the homo-

Al2 In an ideal fluid medium

Only one inhomogeneous plane wave can propagate intoan ideal fluid a lamellar one (curluF = 0) The generalform of the real-valued displacement UF associated with this

514 Favretto-Anres Sessarego Identification of viscoelastic solid parameters

a

ACUSTICA acta acusticaVoIH5 (1999)

a 005 01 015 02 025Distance from the interface (wavelengths)

l

-10

co~gt -20egtQ)c

W

~ -30Olgo

co

~ 0

o 0

c 0

o

(a)

co -10~~ -15Qc

W0-20g -25

f ~~~~~__a 5 10

Distance of propagation (wavelengths)

(a)

15

o

-60

oC 0

(b)

oo

UO

o

-5

co~-1OfgtOlQcw -15oCg~ -20

-25

01 02 03 04 05 06Distance from the interface (wavelength)

-30o 2 4 6 8 10Distance of propagation (wavelength)

12

Figure A6 Differences between theoretical (--) and experimental(0) results quantified in rlB attenuation of the energy of the Stoneley-Scholte wave as a function of distance (a) from the waterPVCinterface frequency 60 kHz wavelength x = 15 em normalizationof energy with the energy obtained for the first position ofthe receiver(06mm from the interface and 2cm from the source) (b) from thewaterB2900 interface frequency 110kHz wavelength x = 12 emnormalization of energy with the energy obtained for the first positionof the receiver (04 mm from the interface and 3 em from the source)

wave is given by equations (A I-A3) Since the medium isnot attenuating equation (A4) then becomes

(A9)

ko = w Co is the wave number of the associated homoge-neous wave Co = Ao Po the longitudinal wave velocity Pothe fluid density and Ao the Lame coefficient Equation (A9)implies that the vectors representing propagation and damp-ing of the inhomogeneous plane wave are perpendicular inthis case the wave is called evanescent wave

A2 The Stoneley-Scholte wave at the plane boundarybetween an ideal fluid and a viscoelastic solid

The propagation of the Stoneley-Scholte wave has been the-oretically analyzed at the plane boundary between an ideal

Figure A7 A comparison between theoretical (--) and experimen-tal (0) results attenuation of the energy of the Stoneley-Scholte waveas a function of distance of propagation at the interface (a) The wa-terPVC interface frequency 75 kHz wavelengthx = 12em (b) thewaterB2900 interface frequency 70 kHz wavelength x = 18emNormalization of energy with the energy obtained for the first po-sition of the receiver (3 em from the source) Differences betweentheoretical and experimental data quantified in dB - - - The slopeobtained by the least squares fit of the experimental data

fluid and a viscoelastic solid [9] The two media were sup-posed to be homogeneous isotropic and semi-infinite Theinterface wave was locally sought in the form of a linearcombination of three inhomogeneous waves (one evanescentwave in the fluid and two heterogeneous waves in the solid)which satisfy boundary conditions at the interface [10]

A21 Theoretical background

One now introduces the perpendicular axes OXl and OX2 withunit vectors Xl and X2 such that the interface lies along OXI

the fluid medium being defined by X2 gt 0 and the solid oneby X2 lt O

The displacement of the lamellar evanescent wave in thefluid and those of the lamellar and torsional heterogeneouswaves in the viscoelastic solid are described in Section A IAll these waves have the same projection K~ of their wavevectors upon the axis Xl The required conditions of con-

ACUSTlCAmiddot acta acusticaVol 85 (1999) FavreUo-Ames Sessarego Identification of viscoelastic solid parameters 515

Table AI Some measured characteristics of the fluid (water) and thesolids (82900 and PVC)

The ideal fluid was represented by water To simulate a vis-coelastic bottom we used either a material called B2900 (it

Frequencies Density LW velocity SW velocity10kHz-I MHz (kgm3) (ms) (ms)

Water 1Ooo 1478PVC 1360 2270 1100

82900 2000 2910 1620

The densities of the energy in the fluid and in the solid as-sociated with the Stoneley-Scholte wave and expressed interms of the components of the strain tensor and its complex-conjugate [10J decrease exponentially from both sides of theinterface Damped propagation takes place along the inter-face with a speed Csch given by

(A22)

Cq is the phase velocity and cg is the group velocity Theyboth depend on the frequency f

Study of the distribution of the acoustical energy in bothmedia (fluid and solid) as a function of the distance fromthe interface showed that 58 of energy associated with theStoneley-Scholte wave is concentrated in the fluid [10] Theenergy does not penetrate deep into the two media (pene-tration length less than one Stoneley-Scholte wavelength)Moreover as the solids are absorbing energy of the inter-face wave is quickly attenuated during its propagation atthe boundary between water and the viscoelastic solids Thelatter remark reveals the difficulty in detecting the Stoneley-Scholte wave at the waterviscoleastic solid interface How-ever as the amount of energy lying in the viscoelastic solidis great one can recover the properties of the solid Thisassumption will be confirmed subsequently

is a synthetic resin which can be loaded with alumina andtungsten and whose velocities are such that Cl gt Ct gt co)or a PVC material (with Ct lt Co lt Cl) The acoustical char-acteristics of the media assumed to be constant in the fre-quency range 10kHz-I MHz are summarized in Table A IThe attenuations of the plane waves in the solid increasewith frequency (see Figures Al and A2) For 500 kHz thelongitudinal wave attenuation was 110 dBm for B2900 and130 dBm for PVC and the shear wave attenuation was180dBm for B2900 and 490dBm for PVC

The numerical solution of equation (AIO) for each fre-quency provided the dispersion curve of the phase velocityof the Stoneley-Scholte wave (see Figure A3) The disper-sion due to the absorbing property of the solid is weak andit takes place in only a small frequency range (10-50 kHz)The group velocity is obtained by numerical differentiationof the phase velocity with respect to frequency [191

f OCq (1)of

Cg (1) == Cq(1) + f oCeP

(1) 1------

Cq(1) of

A23 Experimental verification of numerical results

The purpose of this subsection is to validate the theoreticalresults obtained for the Stoneley-Scholte wave properties bytank experiments In particular it was desired to determinequantitatively the energy damping of the interface wave asa function of propagation path along the boundary betweenwater and viscoelastic solids

The Stoneley-Scholte wave was generated by a solidwedge excited by a shear wave transducer (diameter 28 mmresonance frequency 110kHz) [10J The wedge was madeof Teflonreg a plastic material with weak shear wave velocity(measured velocity 550ms) The angle of the wedge wascalculated to ensure the conversion of an incident shear waveinto an interface wave at the water solid material (PVC orB2900) interface The transmitter and the solid wedge werebonded by a coupling respectively to the Teflonreg materialand to the solid sample (PVC or B2900) The signals were re-ceived by a wideband hydrophone (diameter 1mm) situated

(AI7)

(All)

(A20)(A21)

(AI3)

(AI2)

(A18)(A19)

KL2 == plusmnjkt (1 + iwvL) - K2

KT2 == plusmnVk (1 + iwvt) - K2

K2 == plusmnjk5 - K2

W == 2K2(1- iwvo) + iw(cflc) (vIKL~ + VOK2) - Y (AI4)

Y == k (1 + iwvI) (A15)

X == 2(1- iwvo) + iw(cflc)(vo - VI) (AI6)2 2 2_ 2 cl VI - ct Vt

Vo - 2 2c[ - 2ct

8mKL2KT2 lt 08mKF2 gt O

Csch == wi K~K~ == ~eKn gt max(ko kt kd

with

tinuity at the interface (X2 == 0) that is the slip boundarycondition give the characteristic equation of the Stoneley-Scholte wave in the unknown K This may be put in thefollowing form [9101

(K2 - KT~)W + 2K2 KL2KT2XPo KL2 4 -- kt (1-1WVt) (AlO)P Kp2

A22 Numerical calculations

The terms KM I and KiI2 are the projections of the vectorKM upon the axes Xl and X2 The solution of equation(A I0) in the complex plane K gives the Stoneley-Scholtewave velocity and attenuation for a given frequency Accord-ing to the finite energy principle in both media the branchesfor the different square root functions are selected to be suchthat

516 Favretto-Anres Sessarego Identification of viscoelastic solid parametersACUSTICAmiddot acta acuslica

Vol 85 (1999)

in thc water near the surface of the solid sample (distanceless than Imm) The experiments took place in a tank ofaverage dimensions (25 x I x 1 m3) fitted out with step-ping motors The parallelepipedic samples were immersed inpure water to a depth 300 mm The thicknesses of the sam-ples (PVC 83 mm B2900 78 mm) were large relative to thewavelengths of the Stoneley-Scholte wave (A8PVC = 8 mmA882900 = 12 cm at the frequency 110 kHz) In both ex-periments wideband pulsed waves were produced by a pulsegenerator and the received signals were low-pass filtered inorder to increase the signal to noise ratio (see Figure A4)

Typical waveforms are represented on Figure AS It wasconcluded that the most significant signal detected near thesurface of the solid sample (PVC or B2900) by the receivercorrcsponds to the Stoneley-Scholte wave excited by theshear component of the emitter while the first signal cor-responds to a signal issued from the longitudinal componentof the emitter These facts were entirely verified by mea-suring the velocity of the presumed Stoneley-Scholte waveThe measured value of the velocity Co of sound in waterwas 1478 mls The experimental measurement of the ve-locity C8ch of the Stoneley-Scholtc wave was on average866 plusmn 10 mls for the waterPVC interface and 1260 plusmn 10 mlsfor the waterB2900 interface The difference between the-oretical and experimental velocities of the Stoneley-Scholtewave was below 3 No dispersion ofthe interface wave wasunfortunately observed

The higher the distance from the waterPVC or wa-terB2900 interface the higher the decrease of the energyof the Stoneley-Scholte wave The differences between thetheoretical values and the experimental ones are reported in alogarithmic scale in Figure A6 The curves are normalized tothe energy of the signal observed for the first position of thereceiver The initial distance of the receiver from the wedgewas 2 cm (for PVC) or 3 em (for B2900) and the initial dis-tance ofthe receiver from the interface was 06 mm (for PVC)or 04 mm (for B2900) For each experiment the agreementbetween the measured data and the theoretical prediction wasgood for the whole frequency spectrum The energy of thesignal associated with the Stoneley-Scholte wave also de-creased exponentially during the wave propagation and theinterface wave was completely attenuated at the boundarybetween water and the solid samples at the end of a 15-wavelength propagation as predicted by theory (see FigureA7) These findings thus prove the validity of the theoreticalstudy [9] of the Stoneley-Scholtc wave at the plane boundarybetween an ideal fluid and a viscoelastic solid They are ofgreat interest for study of inverse problcm and they are usedas data for inversion algorithm as it is shown in Section 3

Acknowledgement

The authors would like to gratefully acknowledge one re-viewer for his very kind and very useful critical comments

References

[11 S D Rajan Determination of geoacoustic parameters of theocean bottom-data requirements J Acoust Soc Am 92 (1992)2126-2140

[21 L Bj0m0 J S Papadakis P J Papadakis J Sageloli J-PSessarego S Sun M 1 Taroudakis Identification of seabeddata from acoustic reflections theory and experiment ActaAcustica 2 (1994) 359-374

[3] R Chapman S Levy J Cabrera K Stinson D OldenburgThe estimation of the density P-wave and S-wave speeds ofthe top most layer of sediments from water bottom reflectionarrivals - In Ocean Seismo-Acoustics T Aka J M Berkson(eds) Plenum Press New York 1986703-710

[4] A Caiti T Akal R D Stoll Determination of shear velocityprofiles by inversion of interface wave data - In Shear Wavesin Marine Sediments J M Hovem M D Richardson R DStoll (eds) Kluwer Academic Publishers Dordreeht 1991557-565

[5] B H Ali M K Broadhead Shear wave properties from inver-sion of Scholte wave data -In Full Field Inversion Methods inOcean and Seismo-Acoustics O Diachok A Caiti PGerstoftH Schmidt (eds) KJuwer Academic Publishers Dordrecht1995371-376

[6] T Akal H Schmidt P Curzi The use of Love waves todetermine the geoacoustic properties of marine sediments -In Ocean Seismo-Acoustics T Akal J M Berkson (cds)Plenum Press New York 1986841-852

[7] S D Rajan C S Howitt Determination of the sediment shearspeed profiles from phase and group velocity dispersion dataof SH wave - In Shear Waves in Marine Sediments 1 MHovem M D Richardson R D Stoll (eds) KJuwer Aca-demic Publishers Dordrecht 1991529-536

[8] F B Jensen H Schmidt Shear properties of ocean sedimentsdetermined from numerical modelling of Scholte wave data -In Ocean Seismo-Acoustics T Akal J M Berkson (eds)Plenum Press New York 1986 683-692

[9] N Favretto-Anres Theoretical study of the Stoneley-Scholtewave at the interCace between an ideal fluid and a viscoelasticsolid Acustica Acta Acustica 82 (1996) 829-838

[10] N Favretto-Anres G Rabau Excitation of the Stone ley-Scholte wave at the boundary between an ideal fluid and aviscoelastic solid Journal of Sound and Vibration 203 (1997)193-208

[II] N Favretto-Anres Utilisation des ondes de type Stoneley-Scholte et Love pour la caracterisation acoustique des sedi-ments marins These de IUniversite de la Mediterranee 1997

[12] A Caiti T Akal R D Stoll Estimation of shear wave ve-locity in shallow marine sediments IEEE Journal of OceanicEngineering 19 (1994) 58-72

[13J w H Press S A Teukolsky W T Vetterling B P FlanneryMinimization or maximization of Cunction - In Numerical Re-cipies in Fortran The Art of Scientific Computing UniversityPress Cambridge 1992 390-398

[14] J Guilbot F Magand Determination of the geoacoustical pa-rameters of a sedimentary layer from surface acoustic wavesa neural network approach - In Full Field Inversion Meth-ods in Ocean and Seismo-Acoustics O Diachok A CaitiP Gerstoft H Schmidt (eds) Kluwer Academic PublishersDordrecht 1995 171-176

[15] A Tarantola Inverse problem theory Methods for data fittingand model parameter estimation Elsevier Science PublishersNew York 1987

[16] P D Wasserman Neural computing - theory and practice VanNostrand Reinhold New York 1989

[17] G Caviglia A Morro E Pagani Inhomogeneous waves inviscoelastic media Wave Motion 12 (1990) 143-159

[18] B Poiree Complex harmonic plane waves - In PhysicalAcoustics O Leroy M A Breazeale (cds) Plenum PressNew York 1991

[19] J Guilbot Celerites de phase et de groupe relations explicitesentre les lois de dispersion J d AcoustiqueS (1992) 635-638

506 Favretto-Anres Sessarego Identification of viscoelastic solid parametersACUSTlCA acta acustica

Vol 85 (IYYY

1050

1000(a)

~~

-_r_ bullbull_~~ _ ~ bull ~

~ --------o

00

===CO~COalXlX COX1corrmrruro=

f

160

160

140

140

120

120

L_-------1-_



40

40

20

20

800 bull

o

850

1250

1150

1100 0

1200

1500 I

1~1350 -

C81300 bull

~

150

60 80 100 120 140 160Frequency (kHz)

50 100Frequency (kHz)

40200

1500l

1450

1400f

sect1350sect

-S 1300

~1250

1200

1150r

11000

Figure I Effect of the changes in the shear wave velocity of theviscoelastic medium on the dispersion curves of the phase velocityof the Stoneley-Scholte wave (a) At the waterPVC interface --dispersion curves corresponding to the exact values of the shear wavevelocity of PVC material dispersion curves corresponding to adecrease of the shear wave velocity by 15 0000 dispersion curvescorresponding to an increase of the shear wave velocity by 15 (b)At the waterfB2900 interface dispersion curves correspondingto the exact values of the shear wave velocity of B2900 material0000 dispersion curves corresponding to a decrease of the shearwave velocity by 15 dispersion curves corresponding to anincrease of the shear wave velocity by 15

Figure 2 Effect of the changes in the shear wave attenuation of theviscoelastic medium on the dispersion curves of the phase veloc-ity of the Stoneley-Scholte wave (a) At the waterPVC interface dispersion curves corresponding to the exact values of the shearwave attenuation of PVC material - - dispersion curves corre-sponding to a decrease of the shear wave attenuation by 15 - - -dispersion curves corresponding to an increase of the shear waveattenuation by 15 (b) At the waterfB2900 interface -- dis-persion curves corresponding to the exact values of the shear waveattenuation of B2900 material dispersion curves correspondingto a decrease of the shear wave attenuation by 15 0000 dispersioncurves corresponding to an increase of the shear wave attenuationby 15

Before solving the inverse problem that is the determina-tion of the shear wave parameters of the viscoelastic solidsfrom the propagation characteristics of the interface wave aparametric study of the influence of the physical propertiesof solids on these characteristics was necessary Section 2presents the results of this parametric study These findingswere then used to develop an inversion strategy

Finally Section 3 is devoted to the solution of the inverseproblem Inversion of interface wave data was carried outusing Brents algorithm The inverse scheme was able torecover the shear wave velocity and attenuation of the bot-tom assuming an a priori knowledge of the environment Italso assumed horizontally layered media (plane and paral-lel interfaces) and homogeneous layers Several test resultsare presented in this section The originality of the study

is that the inverse algorithm applied to either dispersion orenergy curves was tested against both noisy synthetic dataand data obtainedfrom experiments performed in laboratoryThe observation and the discussion of differences betweentheoretical and experimental results in the latter case maythus be a good criterion for the validity and applicability ofthe scheme performed under in situ conditions

2 Introduction to inverse problem

It is evident from previous works on forward modeling[9 10 ] 1] that the parameters of the shear waves in vis-coelastic medium conditione the propagation characteristicsof the Stoneley- Scholte wave at the ideal fluidviscoelastic


(c)


1614

(d)

oX


4

100r 100

90 - 90(a)

80 80

70 70

i 60 60gt gtf 50 ~50lt= lt=w w40 40

30 30

20 20

10 x 10 0



l -~--~ lOT90 90

(b)80 0 80 ~ x

I x70 70

60 160gt gt~ 50 ~ 50lt= lt=w w

40 r40

30 30 -

20 20 -

10 10 -

0 --~--- ~ 00 2 4 6 8 10 12 14 16 0 2













90 90(a) (c)

80 80

70 70

~ 60 ~ 60gt gt~ 50 ~ 50l l

W w40 40

30 30 -

20 20



1001 ------- --------------------------- 100

90 - 90 -(b) (d)

80 80

70 70

160 if 60gt gt~ 50 ~ 50l lW W 0

40 40

30 30

20 20 u

10 10

01 ~~ -------L bull 00 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16





3 Inverse problem



31 Some definitions










(2)

410

lO57

1547

45

estimate of Ct (ms)

estimation errors

estimate of C (ms)

estimation errors

1 N 2


B2900

PVC






(1)

---bull~ Passive -


OS k S N







Vol H5 (IW91








PVC 8 88 08 88 0

B2900 3 71 253 65 14

8 86 518 65 14

II 82 44II 65 14

21 9] 6021 65 14

40 50 ]240 65 14










Estimationerrors

Estimate of at(dBm)






4 Conclusion




Vol S5 (] 9991

300

250r (a)

I



50

0 -- -a 100 200 300 400 500 600 700

Frequency (kHz)

400 - --~

350(b)

300E

~250 yflt=0 G) I

~ 200 y- Jlt= E--

_p~_ftri 150 ~

i100

--f

50 ------- I00 100 200 300 400 500 600 700

Frequency (kHz)

700

700

600

600



200

200

100

(b)

00-- 100

20 -

200 J~

100 __-------

E140iil~120t 100

j 80~

60140 -

900

1--800

700t

I 600m~a 500~JIi 400i

300









with

U~f

KrvI

K~l K~f

U~1exp(iK~l r)


2lVI + Aper

(AI)

(A2)

(A3)(A4)

(A5)


950

940~~

1930

gt 920~2910sect~~ 900c0-

890

880

870

0

1440-


120

(a)

140 160

0025(a)

002

0015

~001

~ 0005

~laquo

05 15 2 25 3 35 45Ti me (second)

x 10


1420

1400-

6 1380-

~gt 13601g~ 1340

-0

sect 1320~f l300

1280

126W

12401~ _o 20 40 60 80 100

Frequency (kHz)120

(b)

140 160

~t-005

05 15 2 25T I me (second$)

(b)

35

I

x 10



(A7)

(A8)


wkt(w)





Tank

Receiverllll=l mmgt

ater


Tetlowwedge+

Shear wavetramducer

(A6)






a



l

-10

co~gt -20egtQ)c

W

~ -30Olgo

co

~ 0

o 0

c 0

o

(a)

co -10~~ -15Qc

W0-20g -25

f ~~~~~__a 5 10


(a)

15

o

-60

oC 0

(b)

oo

UO

o

-5


-25



12



(A9)














Water 1Ooo 1478PVC 1360 2270 1100

82900 2000 2910 1620


(A22)





f OCq (1)of

Cg (1) == Cq(1) + f oCeP

(1) 1------

Cq(1) of




(AI7)

(All)

(A20)(A21)

(AI3)

(AI2)

(A18)(A19)





Y == k (1 + iwvI) (A15)


Vo - 2 2c[ - 2ct



with






Vol 85 (1999)




Acknowledgement


References





















(c)


1614

(d)

oX


4

100r 100

90 - 90(a)

80 80

70 70

i 60 60gt gtf 50 ~50lt= lt=w w40 40

30 30

20 20

10 x 10 0



l -~--~ lOT90 90

(b)80 0 80 ~ x

I x70 70

60 160gt gt~ 50 ~ 50lt= lt=w w

40 r40

30 30 -

20 20 -

10 10 -

0 --~--- ~ 00 2 4 6 8 10 12 14 16 0 2













90 90(a) (c)

80 80

70 70

~ 60 ~ 60gt gt~ 50 ~ 50l l

W w40 40

30 30 -

20 20



1001 ------- --------------------------- 100

90 - 90 -(b) (d)

80 80

70 70

160 if 60gt gt~ 50 ~ 50l lW W 0

40 40

30 30

20 20 u

10 10

01 ~~ -------L bull 00 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16





3 Inverse problem



31 Some definitions










(2)

410

lO57

1547

45

estimate of Ct (ms)

estimation errors

estimate of C (ms)

estimation errors

1 N 2


B2900

PVC






(1)

---bull~ Passive -


OS k S N







Vol H5 (IW91








PVC 8 88 08 88 0

B2900 3 71 253 65 14

8 86 518 65 14

II 82 44II 65 14

21 9] 6021 65 14

40 50 ]240 65 14










Estimationerrors

Estimate of at(dBm)






4 Conclusion




Vol S5 (] 9991

300

250r (a)

I



50

0 -- -a 100 200 300 400 500 600 700

Frequency (kHz)

400 - --~

350(b)

300E

~250 yflt=0 G) I

~ 200 y- Jlt= E--

_p~_ftri 150 ~

i100

--f

50 ------- I00 100 200 300 400 500 600 700

Frequency (kHz)

700

700

600

600



200

200

100

(b)

00-- 100

20 -

200 J~

100 __-------

E140iil~120t 100

j 80~

60140 -

900

1--800

700t

I 600m~a 500~JIi 400i

300









with

U~f

KrvI

K~l K~f

U~1exp(iK~l r)


2lVI + Aper

(AI)

(A2)

(A3)(A4)

(A5)


950

940~~

1930

gt 920~2910sect~~ 900c0-

890

880

870

0

1440-


120

(a)

140 160

0025(a)

002

0015

~001

~ 0005

~laquo

05 15 2 25 3 35 45Ti me (second)

x 10


1420

1400-

6 1380-

~gt 13601g~ 1340

-0

sect 1320~f l300

1280

126W

12401~ _o 20 40 60 80 100

Frequency (kHz)120

(b)

140 160

~t-005

05 15 2 25T I me (second$)

(b)

35

I

x 10



(A7)

(A8)


wkt(w)





Tank

Receiverllll=l mmgt

ater


Tetlowwedge+

Shear wavetramducer

(A6)






a



l

-10

co~gt -20egtQ)c

W

~ -30Olgo

co

~ 0

o 0

c 0

o

(a)

co -10~~ -15Qc

W0-20g -25

f ~~~~~__a 5 10


(a)

15

o

-60

oC 0

(b)

oo

UO

o

-5


-25



12



(A9)














Water 1Ooo 1478PVC 1360 2270 1100

82900 2000 2910 1620


(A22)





f OCq (1)of

Cg (1) == Cq(1) + f oCeP

(1) 1------

Cq(1) of




(AI7)

(All)

(A20)(A21)

(AI3)

(AI2)

(A18)(A19)





Y == k (1 + iwvI) (A15)


Vo - 2 2c[ - 2ct



with






Vol 85 (1999)




Acknowledgement


References






















90 90(a) (c)

80 80

70 70

~ 60 ~ 60gt gt~ 50 ~ 50l l

W w40 40

30 30 -

20 20



1001 ------- --------------------------- 100

90 - 90 -(b) (d)

80 80

70 70

160 if 60gt gt~ 50 ~ 50l lW W 0

40 40

30 30

20 20 u

10 10

01 ~~ -------L bull 00 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16





3 Inverse problem



31 Some definitions










(2)

410

lO57

1547

45

estimate of Ct (ms)

estimation errors

estimate of C (ms)

estimation errors

1 N 2


B2900

PVC






(1)

---bull~ Passive -


OS k S N







Vol H5 (IW91








PVC 8 88 08 88 0

B2900 3 71 253 65 14

8 86 518 65 14

II 82 44II 65 14

21 9] 6021 65 14

40 50 ]240 65 14










Estimationerrors

Estimate of at(dBm)






4 Conclusion




Vol S5 (] 9991

300

250r (a)

I



50

0 -- -a 100 200 300 400 500 600 700

Frequency (kHz)

400 - --~

350(b)

300E

~250 yflt=0 G) I

~ 200 y- Jlt= E--

_p~_ftri 150 ~

i100

--f

50 ------- I00 100 200 300 400 500 600 700

Frequency (kHz)

700

700

600

600



200

200

100

(b)

00-- 100

20 -

200 J~

100 __-------

E140iil~120t 100

j 80~

60140 -

900

1--800

700t

I 600m~a 500~JIi 400i

300









with

U~f

KrvI

K~l K~f

U~1exp(iK~l r)


2lVI + Aper

(AI)

(A2)

(A3)(A4)

(A5)


950

940~~

1930

gt 920~2910sect~~ 900c0-

890

880

870

0

1440-


120

(a)

140 160

0025(a)

002

0015

~001

~ 0005

~laquo

05 15 2 25 3 35 45Ti me (second)

x 10


1420

1400-

6 1380-

~gt 13601g~ 1340

-0

sect 1320~f l300

1280

126W

12401~ _o 20 40 60 80 100

Frequency (kHz)120

(b)

140 160

~t-005

05 15 2 25T I me (second$)

(b)

35

I

x 10



(A7)

(A8)


wkt(w)





Tank

Receiverllll=l mmgt

ater


Tetlowwedge+

Shear wavetramducer

(A6)






a



l

-10

co~gt -20egtQ)c

W

~ -30Olgo

co

~ 0

o 0

c 0

o

(a)

co -10~~ -15Qc

W0-20g -25

f ~~~~~__a 5 10


(a)

15

o

-60

oC 0

(b)

oo

UO

o

-5


-25



12



(A9)














Water 1Ooo 1478PVC 1360 2270 1100

82900 2000 2910 1620


(A22)





f OCq (1)of

Cg (1) == Cq(1) + f oCeP

(1) 1------

Cq(1) of




(AI7)

(All)

(A20)(A21)

(AI3)

(AI2)

(A18)(A19)





Y == k (1 + iwvI) (A15)


Vo - 2 2c[ - 2ct



with






Vol 85 (1999)




Acknowledgement


References




























(2)

410

lO57

1547

45

estimate of Ct (ms)

estimation errors

estimate of C (ms)

estimation errors

1 N 2


B2900

PVC






(1)

---bull~ Passive -


OS k S N







Vol H5 (IW91








PVC 8 88 08 88 0

B2900 3 71 253 65 14

8 86 518 65 14

II 82 44II 65 14

21 9] 6021 65 14

40 50 ]240 65 14










Estimationerrors

Estimate of at(dBm)






4 Conclusion




Vol S5 (] 9991

300

250r (a)

I



50

0 -- -a 100 200 300 400 500 600 700

Frequency (kHz)

400 - --~

350(b)

300E

~250 yflt=0 G) I

~ 200 y- Jlt= E--

_p~_ftri 150 ~

i100

--f

50 ------- I00 100 200 300 400 500 600 700

Frequency (kHz)

700

700

600

600



200

200

100

(b)

00-- 100

20 -

200 J~

100 __-------

E140iil~120t 100

j 80~

60140 -

900

1--800

700t

I 600m~a 500~JIi 400i

300









with

U~f

KrvI

K~l K~f

U~1exp(iK~l r)


2lVI + Aper

(AI)

(A2)

(A3)(A4)

(A5)


950

940~~

1930

gt 920~2910sect~~ 900c0-

890

880

870

0

1440-


120

(a)

140 160

0025(a)

002

0015

~001

~ 0005

~laquo

05 15 2 25 3 35 45Ti me (second)

x 10


1420

1400-

6 1380-

~gt 13601g~ 1340

-0

sect 1320~f l300

1280

126W

12401~ _o 20 40 60 80 100

Frequency (kHz)120

(b)

140 160

~t-005

05 15 2 25T I me (second$)

(b)

35

I

x 10



(A7)

(A8)


wkt(w)





Tank

Receiverllll=l mmgt

ater


Tetlowwedge+

Shear wavetramducer

(A6)






a



l

-10

co~gt -20egtQ)c

W

~ -30Olgo

co

~ 0

o 0

c 0

o

(a)

co -10~~ -15Qc

W0-20g -25

f ~~~~~__a 5 10


(a)

15

o

-60

oC 0

(b)

oo

UO

o

-5


-25



12



(A9)














Water 1Ooo 1478PVC 1360 2270 1100

82900 2000 2910 1620


(A22)





f OCq (1)of

Cg (1) == Cq(1) + f oCeP

(1) 1------

Cq(1) of




(AI7)

(All)

(A20)(A21)

(AI3)

(AI2)

(A18)(A19)





Y == k (1 + iwvI) (A15)


Vo - 2 2c[ - 2ct



with






Vol 85 (1999)




Acknowledgement


References





















Vol H5 (IW91








PVC 8 88 08 88 0

B2900 3 71 253 65 14

8 86 518 65 14

II 82 44II 65 14

21 9] 6021 65 14

40 50 ]240 65 14










Estimationerrors

Estimate of at(dBm)






4 Conclusion




Vol S5 (] 9991

300

250r (a)

I



50

0 -- -a 100 200 300 400 500 600 700

Frequency (kHz)

400 - --~

350(b)

300E

~250 yflt=0 G) I

~ 200 y- Jlt= E--

_p~_ftri 150 ~

i100

--f

50 ------- I00 100 200 300 400 500 600 700

Frequency (kHz)

700

700

600

600



200

200

100

(b)

00-- 100

20 -

200 J~

100 __-------

E140iil~120t 100

j 80~

60140 -

900

1--800

700t

I 600m~a 500~JIi 400i

300









with

U~f

KrvI

K~l K~f

U~1exp(iK~l r)


2lVI + Aper

(AI)

(A2)

(A3)(A4)

(A5)


950

940~~

1930

gt 920~2910sect~~ 900c0-

890

880

870

0

1440-


120

(a)

140 160

0025(a)

002

0015

~001

~ 0005

~laquo

05 15 2 25 3 35 45Ti me (second)

x 10


1420

1400-

6 1380-

~gt 13601g~ 1340

-0

sect 1320~f l300

1280

126W

12401~ _o 20 40 60 80 100

Frequency (kHz)120

(b)

140 160

~t-005

05 15 2 25T I me (second$)

(b)

35

I

x 10



(A7)

(A8)


wkt(w)





Tank

Receiverllll=l mmgt

ater


Tetlowwedge+

Shear wavetramducer

(A6)






a



l

-10

co~gt -20egtQ)c

W

~ -30Olgo

co

~ 0

o 0

c 0

o

(a)

co -10~~ -15Qc

W0-20g -25

f ~~~~~__a 5 10


(a)

15

o

-60

oC 0

(b)

oo

UO

o

-5


-25



12



(A9)














Water 1Ooo 1478PVC 1360 2270 1100

82900 2000 2910 1620


(A22)





f OCq (1)of

Cg (1) == Cq(1) + f oCeP

(1) 1------

Cq(1) of




(AI7)

(All)

(A20)(A21)

(AI3)

(AI2)

(A18)(A19)





Y == k (1 + iwvI) (A15)


Vo - 2 2c[ - 2ct



with






Vol 85 (1999)




Acknowledgement


References
























4 Conclusion




Vol S5 (] 9991

300

250r (a)

I



50

0 -- -a 100 200 300 400 500 600 700

Frequency (kHz)

400 - --~

350(b)

300E

~250 yflt=0 G) I

~ 200 y- Jlt= E--

_p~_ftri 150 ~

i100

--f

50 ------- I00 100 200 300 400 500 600 700

Frequency (kHz)

700

700

600

600



200

200

100

(b)

00-- 100

20 -

200 J~

100 __-------

E140iil~120t 100

j 80~

60140 -

900

1--800

700t

I 600m~a 500~JIi 400i

300









with

U~f

KrvI

K~l K~f

U~1exp(iK~l r)


2lVI + Aper

(AI)

(A2)

(A3)(A4)

(A5)


950

940~~

1930

gt 920~2910sect~~ 900c0-

890

880

870

0

1440-


120

(a)

140 160

0025(a)

002

0015

~001

~ 0005

~laquo

05 15 2 25 3 35 45Ti me (second)

x 10


1420

1400-

6 1380-

~gt 13601g~ 1340

-0

sect 1320~f l300

1280

126W

12401~ _o 20 40 60 80 100

Frequency (kHz)120

(b)

140 160

~t-005

05 15 2 25T I me (second$)

(b)

35

I

x 10



(A7)

(A8)


wkt(w)





Tank

Receiverllll=l mmgt

ater


Tetlowwedge+

Shear wavetramducer

(A6)






a



l

-10

co~gt -20egtQ)c

W

~ -30Olgo

co

~ 0

o 0

c 0

o

(a)

co -10~~ -15Qc

W0-20g -25

f ~~~~~__a 5 10


(a)

15

o

-60

oC 0

(b)

oo

UO

o

-5


-25



12



(A9)














Water 1Ooo 1478PVC 1360 2270 1100

82900 2000 2910 1620


(A22)





f OCq (1)of

Cg (1) == Cq(1) + f oCeP

(1) 1------

Cq(1) of




(AI7)

(All)

(A20)(A21)

(AI3)

(AI2)

(A18)(A19)





Y == k (1 + iwvI) (A15)


Vo - 2 2c[ - 2ct



with






Vol 85 (1999)




Acknowledgement


References





















Vol S5 (] 9991

300

250r (a)

I



50

0 -- -a 100 200 300 400 500 600 700

Frequency (kHz)

400 - --~

350(b)

300E

~250 yflt=0 G) I

~ 200 y- Jlt= E--

_p~_ftri 150 ~

i100

--f

50 ------- I00 100 200 300 400 500 600 700

Frequency (kHz)

700

700

600

600



200

200

100

(b)

00-- 100

20 -

200 J~

100 __-------

E140iil~120t 100

j 80~

60140 -

900

1--800

700t

I 600m~a 500~JIi 400i

300









with

U~f

KrvI

K~l K~f

U~1exp(iK~l r)


2lVI + Aper

(AI)

(A2)

(A3)(A4)

(A5)


950

940~~

1930

gt 920~2910sect~~ 900c0-

890

880

870

0

1440-


120

(a)

140 160

0025(a)

002

0015

~001

~ 0005

~laquo

05 15 2 25 3 35 45Ti me (second)

x 10


1420

1400-

6 1380-

~gt 13601g~ 1340

-0

sect 1320~f l300

1280

126W

12401~ _o 20 40 60 80 100

Frequency (kHz)120

(b)

140 160

~t-005

05 15 2 25T I me (second$)

(b)

35

I

x 10



(A7)

(A8)


wkt(w)





Tank

Receiverllll=l mmgt

ater


Tetlowwedge+

Shear wavetramducer

(A6)






a



l

-10

co~gt -20egtQ)c

W

~ -30Olgo

co

~ 0

o 0

c 0

o

(a)

co -10~~ -15Qc

W0-20g -25

f ~~~~~__a 5 10


(a)

15

o

-60

oC 0

(b)

oo

UO

o

-5


-25



12



(A9)














Water 1Ooo 1478PVC 1360 2270 1100

82900 2000 2910 1620


(A22)





f OCq (1)of

Cg (1) == Cq(1) + f oCeP

(1) 1------

Cq(1) of




(AI7)

(All)

(A20)(A21)

(AI3)

(AI2)

(A18)(A19)





Y == k (1 + iwvI) (A15)


Vo - 2 2c[ - 2ct



with






Vol 85 (1999)




Acknowledgement


References





















950

940~~

1930

gt 920~2910sect~~ 900c0-

890

880

870

0

1440-


120

(a)

140 160

0025(a)

002

0015

~001

~ 0005

~laquo

05 15 2 25 3 35 45Ti me (second)

x 10


1420

1400-

6 1380-

~gt 13601g~ 1340

-0

sect 1320~f l300

1280

126W

12401~ _o 20 40 60 80 100

Frequency (kHz)120

(b)

140 160

~t-005

05 15 2 25T I me (second$)

(b)

35

I

x 10



(A7)

(A8)


wkt(w)





Tank

Receiverllll=l mmgt

ater


Tetlowwedge+

Shear wavetramducer

(A6)






a



l

-10

co~gt -20egtQ)c

W

~ -30Olgo

co

~ 0

o 0

c 0

o

(a)

co -10~~ -15Qc

W0-20g -25

f ~~~~~__a 5 10


(a)

15

o

-60

oC 0

(b)

oo

UO

o

-5


-25



12



(A9)














Water 1Ooo 1478PVC 1360 2270 1100

82900 2000 2910 1620


(A22)





f OCq (1)of

Cg (1) == Cq(1) + f oCeP

(1) 1------

Cq(1) of




(AI7)

(All)

(A20)(A21)

(AI3)

(AI2)

(A18)(A19)





Y == k (1 + iwvI) (A15)


Vo - 2 2c[ - 2ct



with






Vol 85 (1999)




Acknowledgement


References





















a



l

-10

co~gt -20egtQ)c

W

~ -30Olgo

co

~ 0

o 0

c 0

o

(a)

co -10~~ -15Qc

W0-20g -25

f ~~~~~__a 5 10


(a)

15

o

-60

oC 0

(b)

oo

UO

o

-5


-25



12



(A9)














Water 1Ooo 1478PVC 1360 2270 1100

82900 2000 2910 1620


(A22)





f OCq (1)of

Cg (1) == Cq(1) + f oCeP

(1) 1------

Cq(1) of




(AI7)

(All)

(A20)(A21)

(AI3)

(AI2)

(A18)(A19)





Y == k (1 + iwvI) (A15)


Vo - 2 2c[ - 2ct



with






Vol 85 (1999)




Acknowledgement


References
























Water 1Ooo 1478PVC 1360 2270 1100

82900 2000 2910 1620


(A22)





f OCq (1)of

Cg (1) == Cq(1) + f oCeP

(1) 1------

Cq(1) of




(AI7)

(All)

(A20)(A21)

(AI3)

(AI2)

(A18)(A19)





Y == k (1 + iwvI) (A15)


Vo - 2 2c[ - 2ct



with






Vol 85 (1999)




Acknowledgement


References





















Vol 85 (1999)




Acknowledgement


References




















inubit - bosch software innovations

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