Determination of Ultrasonic Parameters Based on Attenuation and Dispersion Measurements

Transcription

1 ULTRASOIC IMAGIG, Determnaton o Ultrasonc Parameters Based on Attenaton and Dsperson Measrements PIG HE Department o Bomedcal and Hman Factors Engneerng Wrght State Unersty Dayton, Oho e-mal: phe@cs.wrght.ed In the measrement o acostc attenaton that obeys a power-law α n, the tradtonal throghtransmsson method ses only the ampltde normaton o the recorded plses to determne the two parameters, and n. In ths paper, we propose a new method that tlzes both the ampltde and phase normaton o the plses to determne the two parameters. Accordng to ths method, the two parameters are estmated by smltaneosly perormng a least sqares t to the attenaton data that are dered rom the ampltde spectra o the plses, and to the dsperson data that are dered rom the phase spectra o the plses. By lly tlzng the normaton contaned n the recorded plses and mposng addtonal constrants on the two parameters, the estmaton ncertanty can be redced. Expermental reslts rom two specmens, one hang a lnear attenaton and one hang a nonlnear attenaton, demonstrate that the new method prodces a moderate arance redcton n the case o lnear attenaton, and a sgncant arance redcton n the case o nonlnear attenaton. KEY WORDS: Attenaton; dsperson; least sqares cre ttng; parameter estmaton.. ITRODUCTIO The measrement o acostc attenaton has mportant applcatons n ltrasond tsse characterzaton and nondestrcte materal testng. - For a wde arety o materals, ncldng sot tsses, the attenaton ncreases wth reqency accordng to a power-law relaton: 3,4 α n where and n n are two materal-dependent parameters. Based on sch a power-law relaton, the measrement o acostc attenaton redces to the determnaton o the two parameters: and n. In tro measrement o acostc attenaton can be condcted sng ether a narrowband or broadband throgh-transmsson technqe. 4-5 In a typcal broadband measrement, two transdcers, one or transmttng an ltrasond plse and one or receng the plse, are placed n a water tank. A plse s rst recorded when there s only a water path between the two transdcers. The specmen s then nserted n between the two transdcers and a second plse s recorded. From the ampltde spectra o the two plses, the attenaton s rst determned at a nmber o dscrete reqences, and the two parameters, and n, are then estmated by least sqares ttng the measred attenaton wth a power-law cre. Althogh ths wdely sed method s qte smple and reasonably accrate n measrng the attenaton, t s oten dclt to prode a dente ale or each o the two parameters. In some cases, the measred attenaton may be tted almost eqally well wth cres assocated wth derent ales o and n. In other cases, a slght change n the reqency range or cre ttng may reslt n a sgncant change n or/and n. Ultrasond relecton at the wa /99 $8. Copyrght 999 by Dynameda, Inc. All rghts o reprodcton n any orm resered.

2 76 PIG HE ter-specmen nterace may prodce addtonal ncertantes n determnng the nddal ales o and n. In order to eectely redce the estmaton ncertantes, one may hae to ncrease the reqency range or attenaton measrement. In ths paper, we propose a new method or estmatng and n that can eectely redce the estmaton ncertantes wthot ncreasng the reqency range or the measrement. Unlke the tradtonal method that tlzes only the ampltde normaton o the two recorded plses, the new method ses both the ampltde and phase normaton o the recorded plses. Accordng to ths method, the two parameters, and n, are obtaned by smltaneosly perormng a least sqares t to the attenaton data that are dered rom the ampltde spectra o the plses, and to the dsperson data that are dered rom the phase spectra o the plses. By lly tlzng the normaton contaned n the recorded plses and mposng addtonal dsperson constrants on the two parameters, the estmaton ncertanty can be redced. The paper s organzed as ollows. The methods or measrng the attenaton and dsperson sng a broadband, throgh-transmsson technqe are rst reewed. An emphass s lad pon a recently deeloped method or dsperson measrement that s easy to perorm and does not reqre phase nwrappng. 6,7 The models or attenaton and dsperson are then presented. The model or attenaton s based on Eq. and the model or dsperson s based on a tme casal model recently proposed by Szabo. 8 Ater ntrodcng a total sqared error TSE ncton that s a weghted sm o the derence between the measred and modeled attenaton and the derence between the measred and modeled dsperson, the maxmm lkelhood estmates or and n are dered, rst or the case o lnear attenaton and then or the case o nonlnear attenaton. Expermental reslts rom two specmens are then reported. The rst specmen exhbts a lnear attenaton and the second specmen exhbts a nonlnear attenaton. For each specmen, parameters are estmated sng both the tradtonal method and the new method or a nmber o reqency ranges oer whch cre ttng s perormed. The reslts show that when the ttng range s gradally decreased, the estmates obtaned by the new method are more precse than the estmates obtaned by the tradtonal method. The mproement proded by the new method s more dramatc or nonlnear attenaton.. METHOD A. Measrement o attenaton and dsperson Fgre shows the smpled expermental setp or the broadband throgh-transmsson measrement. P w t represents the receed plse wth the water path only, and P s t represents the receed plse when the specmen s nserted. I we neglect the attenaton o the water, the attenaton o the specmen can be ond as: 5 α lnt L A w ln L As where A w and A s are the ampltde spectra o P w t and P s t, respectely; L s the thckness o the specmen; T 4z w z s /z w z s s the combned transmsson coecent at the two water-specmen nteraces assmng normal ncdence, and z w and z s are the acostc mpedance o the water and the specmen, respectely. To measre the dsperson, a method recently proposed by He s sed. 6,7 By drectly measrng the change n phase elocty between a certan reqency and a reerence reqency

3 DETERMIATIO OF ULTRASOIC PARAMETERS 77 FIG. Expermental set-p or the measrement o attenaton and dsperson sng the throgh-transmsson technqe. rather than measrng the absolte phase elocty at any partclar reqency, the method elmnates the need or measrng the trggerng delays o the samplng wndow and does not reqre phase nwrappng whch s assocated wth the annoyng mπ phase ambgty. 9 I we neglect the dsperson o water, the dsperson o the specmen can be ond as: 7 V V p p φ φ φ φ πl πl w s w s 3 where s a reerence reqency where the plses P w t and P s t hae sgncant energy; V p s the phase elocty; φ w and φ s are the phase spectra o P w t and P s t, respectely, ater the center o each plse s crclarly shted to the begnnng o the samplng wndow. In the actal experment, A w and A s are obtaned by Fast Forer Transorm FFT. As a reslt, n the actal calclaton, the arable n Eqs. and 3 wll be replaced by dscrete reqences,, whch coer the entre sel reqency range o P w t and P s t. B. Tradtonal method or determnng and n Method The tradtonal method or determnng and n s based on attenaton measrement only. Rewrte Eq. n the ollowng orm and se dscrete reqences: Aw ln L α α As, 4 where are the measred attenaton data, and α lnt/l. IT s precsely known, then are least sqares tted wth a cre α n, where and n are the only two n-

4 78 PIG HE known parameters. I T s not precsely known, then α becomes the thrd nknown parameter n cre ttng. In ths stdy, we consder the more general case o nknown T. C. Proposed method or determnng and n Method Snce T s generally nknown or s dclt to determne accrately, we dene a new ncton to elmnate the eects o T: where s the same reerence reqency n Eq. 3. Comparng Eq. 5 wth Eq. 3, one notces the symmetrcal way n whch the attenaton and dsperson are measred and expressed. As we wll show later, ths symmetrc orm o and helps to determne an optmal weghtng actor between the attenaton measrement and dsperson measrements n estmatng and n by least sqares ttng. Accordng to the power-law relaton α n, the model or n Eq. 5 s: To model the dsperson ncton n Eq. 3, we adopt a tme casal model deeloped by Szabo. 8 Ths model has been ond qte accrate or both lnear and nonlnear attenaton. 7 Let s rst consder the case o lnear attenaton n. Accordng to Szabo s model, when n, the model or dsperson s: ln Aw ln As ln Aw ln As α α L L n * n 6 * 7 ln ln π The same dsperson was also predcted by the nearly local model proposed earler by O Donnel et al. To estmate based on both the attenaton and dsperson measrements, we dene a total sqared error TSE ncton: [ * ] λ [ * ] TSE λ where, *,, * are dened n Eqs. 5, 6, 3 and 7, respectely. The two parameters, λ and λ, n Eq. 8 hae two roles. Frst o all, they are needed de to the act that and hae derent nts. Secondly, they are ntrodced to dere an optmal weghtng actor between the attenaton and dsperson measrements n estmatng the ale o. Sbstttng the models or attenaton * and dsperson * as dened n Eqs. 6 and 7, and solng or based on the eqaton: TSE, we obtan the optmal maxmm lkelhood estmate o that mnmzes the TSE: 5 8 k ln ln π k ln ln 4 π 9

5 where k λ λ s a new parameter,, and. ow let s nd the optmal ale o k that mnmzes the arance o the estmate. I we se and to represent the mean arance o the attenaton measrement and dsperson measrement, respectely, the arance o the estmate can be obtaned as: where and are the arance o each measrement and, respectely. In derng Eq., we assme and. By solng k, we obtan the optmal ale o k that mnmzes : Snce the actal ales o and V are nknown, we rther assme that and V are proportonal to the mean sqare ales o the measred attenaton and dsperson, respectely: By sbstttng ths ale o k nto Eq. 9, we obtan the optmal estmate o based on the measred attenaton and dsperson: In addton, by sng Eq., we may redene the total sqared error as: DETERMIATIO OF ULTRASOIC PARAMETERS ln ln ln ln k k π π k λ λ k λ λ 4 ln ln ln ln π π 3 [ ] TSE ln ln π 4

6 8 PIG HE For nonlnear attenaton n >, Szabo s tme casal model ges the ollowng expresson o dsperson: 8 * A new total sqared error ncton, smlar to the one n Eq. 4 can then be dened based on the nonlnear attenaton and nonlnear dsperson models: TSE n n [ ] tan nπ / π n n 5 tan nπ / π n In ths case, there are no closed-orm soltons or and n. To nd the optmal estmates o these two parameters that mnmze TSE, one may perorm an terate grd search wth two parameters and n. 3 On the other hand, the speed o the search can be sgncantly ncreased we perorm a one-parameter search. To do ths, we rst show that or any gen n, the optmal ale o that mnmzes TSE n Eq. 6 has a closed-orm solton: n 6 n n tan nπ / n π n n tan nπ / n n 4π n 7 The one-parameter search rotne contans the ollowng steps: specy an ntal ale or n and an ncrement step n; nd sng Eq. 7; calclate TSE n Eq. 6 sng the crrent ales o and n; ncrement n by n, calclate the new ales o and TSE; compare the new ale o TSE wth the preos ale o TSE, decreased, go back to step ; ncreased, let n n; and then go back to step. Ths process contnes ntl TSE no longer changes. The nal ales o and n are the optmal estmates. 3. EXPERIMET AD RESULTS To test the new method, a throgh-transmsson experment s condcted sng two specmens: a cylndrcal block made o Plexglas that exhbts a lnear attenaton, and a phantom made o castor ol that exhbts a nonlnear attenaton. The Plexglas block has a dameter o 6.3 cm and a thckness L o 4 cm. To make the caster ol phantom, the two ends o a plastc tbe nner dameter 7.6 cm, L 5.5 cm are sealed wth thn polyethylene lm, and the tbe s then lled wth castor ol. The transmttng and receng transdcers sed n ths stdy are Panametrcs V38 3.5MHz, 3 mm apertre, 8.9 cm ocal dstance and V MHz, 6.35 mm apertre, nonocsng, respectely. The dstance between the two transdcers s 8 cm, and the water temperatre s C. A Panametrcs 55PR plse/receer s sed to dre the transmttng transdcer and recees the transmtted plse. The otpt rom the receer s dgtzed by a Sony/Tek 39AD programmable dgtzer that has a -bt resolton and a samplng reqency o 6 MHz. Each samplng wndow contans

7 DETERMIATIO OF ULTRASOIC PARAMETERS 8 FIG. Parameter estmaton sng the tradtonal method or the Plexglas specmen. The crcles represent the measred attenaton. The sold lne s the tted least sqares lne n the reqency range o - 6 MHz, and the dashed lne s the tted least sqares lne n the range o 3-4 MHz.,4 samples and the sampled plses are aeraged tmes to mproe the sgnal-to-nose rato. For the Plexglas specmen, the sel reqency range or measrng the attenaton and dsperson s ond to be rom MHz to 6 MHz. The reerence reqency s chosen as the central reqency 3.5 MHz. The attenaton and dsperson are calclated accordng to Eqs. 4, 5 and 3, respectely. The measred attenaton exhbts a lnear reqency dependence. The parameter s rst estmated sng Method the tradtonal method by ttng the measred attenaton,, wth a straght lneα, where α s also an nknown parameter. To test the arablty o the estmate, cre ttng s perormed n or addtonal reqency ranges: MHz, -5 MHz, MHz, and 3-4 MHz. The mean and standard deaton o the e estmates are then calclated. For Method the proposed method, the parameter s calclated sng Eq. 3. Agan, e ales o are obtaned or the aboe e reqency ranges, and the mean and standard deaton are then calclated. For the castor ol phantom, the sel reqency range s ond to be rom MHz to 5 MHz, and a reerence reqency 3 MHz s chosen. The measred attenaton o ths specmen shows a marked nonlnear reqency dependence. As a reslt, nonlnear cre ttng s employed. For Method, the ales o and n are estmated by ttng the measred attenaton,, wth a power creα n whereα, and n are three nknown parameters. For Method, the ales o and n are estmated sng the one-parameter search rotne that ncldes Eq. 7, as descrbed earler. For each method, parameter estmaton s perormed n or derent reqency ranges: -5 MHz, MHz, -4 MHz, and MHz, and the mean and standard deaton o the and n estmates are agan calclated. Fgre shows the estmaton made on the Plexglas specmen sng the tradtonal method that perorms cre ttng sng the attenaton data only. For ths specmen, n and only needs to be estmated. The crcles represent the measred attenaton,, sng Eq. 4. The sold lne s obtaned by perormng least sqares ttng n the reqency range o -6 MHz. The dashed lne s obtaned by perormng least sqares ttng n the reqency range o 3-4 MHz. Fgre 3 shows the estmaton sng the new method that smltaneosly perorms cre ttng on the attenaton data Fg. 3a and the dsperson data Fg. 3b. Agan, the sold lnes are obtaned or the ttng range o -6 MHz and the dashed lnes are obtaned or the ttng range o 3-4 MHz. By comparng gre wth gre 3a, one notces that the sold lne and the dashed lne n gre 3a are closer to each other than the two lnes n gre, ndcatng that the arablty o estmaton sng the new method

8 8 PIG HE FIG. 3 a Parameter estmaton sng the new method or the plexglas specmen. The crcles represent the measred derental attenaton. The reerence reqency s 3.5 MHz. The sold lne [ ] s the smltaneosly tted together wth the cre n gre 3b least sqares lne n the reqency range o -6 MHz, and the dashed lne [.933 ] s the smltaneosly tted least sqares lne n the range o 3-4 Mhz. b The crcles represent the measred dsperson. The reerence reqency s 3.5 MHz. The sold lne [.967 ln -ln ] s the smltaneosly tted together wth the cre n gre 3a least sqares lne n the reqency range o -6 MHz, and the dashed lne [.945 ln ln ] s the tted least sqares lne n the range o 3-4 MHz. TABLE Comparson o two methods n estmatng o the Plexglas specmen. The nt o s db cm - MHz -. Fttng range MHz Method Method mean standard deaton..9

9 DETERMIATIO OF ULTRASOIC PARAMETERS 83 FIG. 4 Parameter estmaton sng the tradtonal method or the castor ol specmen. The crcles represent the measred attenaton. The sold lne s the tted least sqares lne n the reqency range o -5 MHz, and the dashed lne s the smltaneosly tted least sqares lne n the range o MHz. Table Comparson o two methods n estmatng and n o the castor ol specmen. The nt o s db cm - MHz -n. Fttng range MHz Method Method n n mean s.d s smaller. The detaled reslts o estmaton sng the two methods are lsted n table. Method reers to the tradtonal method and Method reers to the proposed method. The table lsts the estmate obtaned n each o the e ttng ranges sng each method. The means o estmates sng the two methods are almost the same.94 s..948, bt the standard deaton o estmates sng the new method.9 s 5% smaller than that o the tradtonal method.. Fgres 4 and 5 show the parameter estmaton perormed on the castor ol phantom, and table lsts the and n estmates obtaned n each o the or ttng ranges sng the two methods. Agan, by comparng gre 4 wth gre 5a, one notces that the sold cre wth ttng range o -5 MHz and the dashed cre wth ttng range o MHz n gre 5a are mch closer to each other than the two cres n gre 4. Accordng to table, the standard deaton o estmate sng Method s tmes smaller than that o Method, and the standard deaton o n estmate s 7 tmes smaller.

10 84 PIG HE FIG. 5 a Parameter estmaton sng the new method or the castor ol specmen. The crcles represent the measred derental attenaton. The reerence reqency s 3 MHz. The sold lne [ ] s the smltaneosly tted together wth the cre n gre 5b least sqares lne n the reqency range o -5 MHz, and the dashed lne [ ] s the smltaneosly tted least sqares lne n the range o MHz. b The crcles represent the measred dsperson. The reerence reqency s 3.5 MHz. The sold lne [ ] s the smltaneosly tted together wth the cre n gre 5a least sqares lne n the reqency range o -5 MHz that corresponds to.7 and n.666, and the dashed lne [ ] s the smltaneosly tted least sqares lne n the range o MHz that corresponds to.686 and n COCLUSIO AD DISCUSSIO A new method s proposed or estmatng the two acostc parameters, and n, that ses both the ampltde and phase normaton o the recorded plses n a throgh-transmsson measrement. Ths new method s blt pon two recent deelopments nolng acostc dsperson: a A tme casal model was deeloped by Szabo 8 that prodes explct and accrate relatons between the dsperson and the two parameter and n, as dened by Eqs. 7 and 5. Together wth Eq., these three eqatons ndcate that and n shold be consdered as two ndamental materal parameters rather than jst two attenaton parameters. Conseqently, determnaton o these two parameters based on both attenaton and dsperson measrement s a more logcal approach. b A method was proposed by He 6,7 that makes the measrement o dsperson rom the phase spectra o the recorded plses n a throgh-transmsson experment as easy as the measrement o attenaton rom the am-

11 DETERMIATIO OF ULTRASOIC PARAMETERS 85 pltde spectra. Becase the phase normaton s readly aalable n a throgh- transmsson experment, the mproement n parameter estmaton s acheed wthot addtonal cost. The man adantage o the proposed method s acheng a arance redcton n estmatng and n. The major premse s that the phase spectra o the transmtted plse prode addtonal normaton, ndependent o that o the ampltde spectra, abot the tre ales o and n. Based on ths premse, we now ge a more qanttate analyss on the arance redcton n estmatng or the case o lnear attenaton. The arance o the estmate sng the tradtonal method based on Eq. 4 can be ormlated as: 8 where s the nmber o ndependent sample ponts and s the arance o each sample. Expand arond a center reqency so that, or, ±, ±,. ±k, where k, and se the relatons:, and -/ 3 /, we obtan B 9 where B s the reqency range sed or cre ttng. Eqaton 9 ndcates that the arance o the estmate can be redced by ncreasng or/and B. A caton abot the nmber s n order. For a partclar par o recorded plses P w t and P s t, the eecte ale o, or the maxmm nmber o ndependent sample ponts, s xed. Althogh the actal nmber o sample ponts sed or cre ttng can be arbtrarly ncreased by rst paddng zeros to the recorded plses and then perormng FFT, sch a process only adds correlated samples bt wll not ncrease the eecte ale o. Snce the added zeros do not contan any normaton abot, the process o zero paddng shold not redce. On the other hand, the phase spectra o the plses contan addtonal normaton abot. By smltaneosly ttng an attenaton cre and a dsperson cre, the dsperson samples whch has the same nmber as the attenaton samples mpose addtonal constrants to the estmate. As a reslt, the eecte ale o s ncreased. Based on ths analyss, the dsperson cre s also a straght lne actally t s not, the new method wold decrease approxmately by a actor o. From table, o the new method s.78 tmes smaller than that o the tradtonal method, ndcatng the actal arance redcton n ths partclar experment s less than, bt close to tmes. For the nonlnear attenaton, the qanttate analyss o arance redcton s rather dclt. From table, o the new method s 4 tmes smaller, and n s 47 tmes smaller than that o the tradtonal method, respectely. These reslts ndcate that the arance redcton n the case o nonlnear attenaton s mch more dramatc. From tables and, one may also notce that o the tradtonal method or the case o nonlnear attenaton s sgncant larger 63 tmes than o the same method or the case o lnear attenaton. An obos reason s that n nonlnear attenaton cre ttng, there are three nknown parameters,, n and α, whle n lnear attenaton cre ttng, there are only two nknown parameters, and α. Some researchers hae tred to x the parameter α by calclatng the ale o the transmsson coecent T n Eq. 4 based on the measrement o the materal propertes. 5 There are two problems wth ths approach. Frstly, t s sometme ery dclt to determne the exact ale o T. For example, the castor ol phantom sed n ths stdy has a thn polyethylene lm that separates the castor ol rom the water.

12 86 PIG HE For sch a water-specmen nterace, T wll be ery dclt to determne de to the presence o ths lm. Secondly, there s an error n determnng T, ths error wll prodce sgncant bas, or systematc error, n and n estmaton. The method proposed n ths stdy achees sgncant arance redcton wthot sch a rsk. It shold be ponted ot that the dscsson so ar abot the arance redcton s based on a random error model or attenaton and dsperson measrements. In the actal measrement, there are many sorces that prodce systematc errors. For example, the reqency response o the receer s not perectly lat wthn the reqency range o nterest, the estmated and n wll be based. Systematc errors can also be prodced by the dracton eects 4,5 and nonlnear propagaton eects. 6,7 There s no general method to analyze and remoe systematc errors. Howeer, the proposed method may stll hae some adantages oer the tradtonal method een n the presence o systematc errors. Ths statement s based on the act that the ampltde spectra and the phase spectra o the transmtted plses are n general aected by the aros error sorces n derent ways. In some cases, the bas o the estmate may be nnotceable when only the attenaton cre s tted, bt may be reealed by the large resdal errors when both the attenaton cre and the dsperson cre are tted smltaneosly. In other cases, the eects o the error sorces on the ampltde spectra and phase spectra may be partally canceled each other so that the nal bas o the estmated parameter s redced. To ery these adantages howeer, more rgoros analyss and carel experments are needed. REFERECES. Bamber, J.C., Hll, C.R., Acostc propertes o normal and canceros hman ler I: Dependence on pathologcal condton, Ultrasond Med. Bol. 7, Klne, R.A., Measrement o attenaton and dsperson sng an ltrasonc spectroscopy technqe, J. Acost. Soc. Am. 76, Hll, C. R. Ultrasonc attenaton and scatterng by tsses, n Handbook o Clncal Ultrasond M. de. Veger et al, eds., pp Wley, ew York, arayana, P.A., Ophr, J., On the aldty o the lnear approxmaton n the parametrc measrement o attenaton n tsses, Ultrasond Med. Bol. 9, Lee, C.C., Lahham, M., and Martn, B.G., Expermental ercaton o the Kramers- Krong relatonshp or acostc waes, IEEE Trans. Ultrason. Ferroelec. Freq. Contr. 37, He, P., On the measrement o ltrasonc dsperson sng a broadband plse technqe abstract, Ultrasonc Imagng, 74, He, P., Drect measrement o ltrasonc dsperson sng a broadband transmsson technqe Ultrasoncs 37, Szabo, T.L., Casal theores and data or acostc attenaton obeyng a reqency power law, J. Acost. Soc. Am. 97, Mobley, J., Marsh, J.., Hall, C.S., Hghes, M.S., Brandenbrger G.H., and Mller, J. G., Broadband measrements o phase elocty n Albnex sspensons, J. Acost. Soc. Am. 4, Rokhln, S.I., Lews, D.K., Gra, K.F., and Adler, L., Real-tme stdy o reqency dependence o attenaton and elocty o ltrasonc waes drng the crng reacton o epoxy resn, J. Acost. Soc. Am. 79, O Donnell, M., Jaynes, E.T., and Mller, J.G., Kramers-Krong relatonshp between ltrasonc attenaton and phase elocty, J. Acost. Soc. Am. 69, Bengton, P.R., Least-sqares t to a straght lne, n Data Redcton and Error Analyss or the Physcal Scences, pp. 9-8 McGraw-Hll, ew York, Bengton, P. R., Least-sqares t to an arbtrary ncton, n Data Redcton and Error Analyss or the Physcal Scences, pp McGraw-Hll, ew York, 969.

13 DETERMIATIO OF ULTRASOIC PARAMETERS X, W., and Kaman, J.J., Dracton correcton methods or nserton ltrasond attenaton estmaton, IEEE Trans. Bomed. Eng. 4, Kaman, J.J., and X, W., Dracton eects n nserton mode estmaton o ltrasonc grop elocty, IEEE Trans. Ultrason. Ferroelect. Freq. Contr. 4, Zeqr, B., Errors n attenaton measrements de to nonlnear propagaton eects, J. Acost. Soc. Am. 9, W, J., Eects o nonlnear nteracton on measrements o reqency-dependent attenaton coecents, J. Acost. Soc. Am. 99,