compressional and shear wave properties of marine sediments

Compressional and shear wave properties of marine sediments:Comparisons between theory and data

Michael J. Buckinghama)

Marine Physical Laboratory, Scripps Institution of Oceanography, University of California, San Diego,9500 Gilman Drive, La Jolla, California 92093-0238

~Received 7 June 2004; accepted for publication 26 August 2004!

According to a recently developed theory of wave propagation in marine sediments, the dispersionrelationships for the phase speed and attenuation of the compressional and the shear wave dependon only three macroscopic physical variables: porosity, grain size, and depth in the sediment. Thedispersion relations also involve three~real! parameters, assigned fixed values, representingmicroscopic processes occurring at grain contacts. The dispersion relationships are compared withextensive data sets, taken from the literature, covering the four wave properties as functions of allthree physical variables. With no adjustable parameters available, the theory matches accurately thetrends of all the data sets. This agreement extends to the compressional and shear attenuations, inthat the theory accurately traces out thelower bound to the widely distributed measuredattenuations: the theory predicts theintrinsic attenuation, arising from the irreversible conversion ofwave energy into heat, whereas the measurements return theeffectiveattenuation, which includesthe intrinsic attenuation plus additional sources of loss such as scattering from shell fragments andother inhomogeneities in the medium. Provided one wave or physical property is known, say thecompressional speed or the porosity, all the remaining sediment properties may be reliably estimatedfrom the theory. ©2005 Acoustical Society of America.@DOI: 10.1121/1.1810231#

PACS numbers: 43.30.Ma@AIT # Pages: 137–152

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I. INTRODUCTION

Over recent decades, extensive data sets have beenlished by Hamiltonet al.1–8 and Richardsonet al.9–12 on thewave properties of surficial, unconsolidated marine sements. It is evident from the data that an unconsolidasediment is capable of supporting two types of propagawave, a compressional~longitudinal! wave and a muchslower shear~transverse! wave. Although there have beeseveral attempts to detect a third type of wave, the ‘‘slocompressional wave of the Biot theory,13,14 all have returneda negative result, including the most recent experimentSimpsonet al.15 Based on this evidence, it is tacitly assumthroughout the following discussion that the slow wave inunconsolidated sediment is negligible if not absent agether.

It is well established from the published data that tphase speed and the attenuation of both the compressand shear wave depend, in a more or less systematic wathe physical properties of the sediment, principally the prosity, the bulk density, the mean grain size, and the debeneath the seafloor. Indeed, both Hamilton3,7,16 andRichardson9,11,12 have developed a set of empirical regresion equations, each one of which expresses a wave prop~e.g., the compressional phase speed! in terms of a physicalproperty of the medium~e.g., the porosity!.

Although regression equations can be satisfactorily fitto the data, thus providing a useful predictive tool, they glittle insight into the physical mechanisms underlying t

a!Also affiliated with the Institute of Sound and Vibration Research, TUniversity, Southampton SO17 1BJ, England. Electronic [email protected]

J. Acoust. Soc. Am. 117 (1), January 2005 0001-4966/2005/117(1)/1

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observed inter-relationships between wave parametersphysical properties. To achieve an understanding of theserved dependencies, it is necessary to turn to a theoremodel of wave propagation in the medium and, in particuto the dispersion relationships predicted by the model. Sa model has recently been developed by Buckingham17 onthe basis of a specific form of dissipation arising at grain-grain contacts. This grain-shearing~G-S! model is intendedto represent wave propagation in an unconsolidated granmedium, that is to say, a material in which the mineral graare in contact but unbonded. By definition, this conditiontaken to mean that the mineral matrix has no intrinstrength or, equivalently, that the elastic~bulk and shear!frame moduli are identically zero.

The absence of an elastic frame in the G-S model ctrasts with the starting assumption in Biot’s classtheory13,14of wave propagation in porous media. Biot treatthe medium as though it possessed an elastic mineral fraan essential assumption in his analysis since the elasticithe means by which a shear wave is supported in his moIn the G-S model, on the other hand, it is not necessarypostulate the existence of an elastic frame because elatype behavior emerges naturally from the analysis as a reof the intergranular interactions themselves: grain-to-grsliding introduces a degree of rigidity into the mediumwhich, amongst other effects, automatically leads to the sport of a shear wave. The lack of elastic~bulk and shear!frame moduli in the G-S model is consistent with the comonplace observations that an unconfined pile of sand grshows no resistance to deformation~indicating no restoringforce! and that sand grains may be picked off a pile~indicat-ing no tensile strength!. Moreover, even though the grain

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remain unbonded, loose sand in a container shows increaresistance to penetration~shear strength!, which can onlyarise from intergranular interactions. Such interactions fothe essence of the G-S wave-propagation model.17

The purpose of this article is to compare the theoretproperties of compressional and shear waves in marine sments, as expressed in the dispersion relationships of themodel, with an extensive set of data culled from the opliterature. Most of the data examined here were obtaifrom in situ measurements in silicilastic sediments.

Besides frequency, the G-S dispersion expressionsthe phase speed and attenuation depend explicitly onmean grain size, the porosity, the bulk density~which isstrongly correlated with the porosity and hence is notindependent variable!, and the overburden pressure~whichtranslates into depth in the sediment!. In addition, the G-Sdispersion relationships involve three unknown constawhich characterize the microscopic processes that occuadjacent grains slide against one another during the pasof a wave. Once the numerical values of these three cstants have been determined, by comparison with three sfrequency data points, all the functional dependencies ofG-S theory may be evaluated and compared with the dThus, the predicted relationships between wave prope~e.g., shear speed! and physical properties~e.g., depth insediment! may be compared directly with the correspondidata sets that have appeared in the literature. Such comsons are examined in some detail in this article, along wthe dispersion curves~i.e., phase speeds and attenuations vsus frequency! predicted by the G-S theory, which are showto match the available data over wide frequency ranges.

II. THE G-S DISPERSION RELATIONS

According to Buckingham,17 the compressional-wavspeed,cp , and attenuation,ap , are given by the expression

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and

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where j 5A21. The corresponding expressions for tshear-wave speed,cs , and attenuation,as , are

cs5Ags

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and

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4 D . ~4!

Between them, Eqs.~1!–~4! constitute the dispersion relationships predicted by the G-S theory. Both dispersion pare causal, satisfying the Kramers–Kronig relationsh

138 J. Acoust. Soc. Am., Vol. 117, No. 1, January 2005

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Several familiar parameters appear in the G-S dispersionpressions: the angular frequency,v, the bulk density of themedium, ro , the sound speed in the absence of grain-grain interactions,co , and an arbitrary timeT51 s, intro-duced solely to avoid awkward dimensions that would oerwise arise when the frequency is raised to a fractiopower.

Less familiar are the three remaining parameters,gp ,gs , and the~positive, fractional! index n, which betweenthem represent the effects of grain-to-grain interactionsthe wave speeds and attenuations. From the way thatappear in the dispersion relations, and the fact that they hdimensions of pressure, it is evident that the two~real! pa-rametersgp andgs are compressional and shear moduli, rspectively, providing a measure of the normal and tangenstresses associated with intergranular sliding. In fact,gp andgs are closely analogous to the Lame´ parameters of elasticitytheory. There is no such analogy, however, for the dimsionless indexn, which is a measure of the degree of strahardening that is postulated to occur at intergranular contas grain-to-grain sliding progresses. Details of the intgranular sliding and strain-hardening mechanisms mayfound in Buckingham.17

If n were zero, the compressional and shear attenuatwould both vanish and the expressions for the two waspeeds would be independent of frequency. Of course, dpation is never completely absent, son is always finite, but itis small compared with unity, taking a value close to 0.1a typical sand sediment. With such a low value forn, the twoG-S expressions for the wave speeds exhibit logarithmicpersion, at levels of the order of 1%~compressional! and10% ~shear! per decade of frequency, and the associatedtenuations both scale essentially as the first power ofquency,f. These simple frequency-dependencies deriverectly from straightforward approximations17 for the exactexpressions in Eqs.~1!–~4!.

From the point of view of wave propagation, two impotant physical properties of sediments are the porosity,N, andthe mean grain diameter,ug . In addition, the intergranulainteractions, and hence also the wave properties, are sensto the overburden pressure, which scales with the depth,d, inthe sediment. All three parameters appear in the G-S dission relations, the porosity throughco and ro , while thegrain size and the depth in the sediment, both raised to ftional powers, appear in the expressions given below forgp

andgs . These two moduli also show a weak but significadependence on the porosity,N.

It is well known that the bulk density,ro , may be ex-pressed in terms of the porosity as a weighted mean ofdensity of the pore water,rw and the density of the mineragrains,rg

ro5Nrw1~12N!rg . ~5!

Similarly, the bulk modulus of the medium,ko , may beexpressed in terms of the porosity as a weighted mean

1

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Michael J. Buckingham: Wave properties of marine sediments

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wherekw andkg are the bulk moduli of the pore water anthe mineral grains, respectively. If the sediment weresimple suspension in which grain-to-grain interactions wabsent, the speed of sound would beco , which depends onthe porosity through Wood’s equation18

co5Ako

ro. ~7!

In the limit of low frequency, the G-S expression for thcompressional wave speed in Eq.~1! reduces toco , whileEqs. ~2!–~4! show thatap , cs , and as all approach zero.Thus, according to the G-S dispersion relations, in the lofrequency limit, the effects of grain-to-grain interactions anegligible and the medium acts as a simple suspension.

Turning to the compressional and shear moduli,gp andgs , their dependencies on grain size, depth in the sedimand porosity are established by treating the mineral grainelastic spheres, which deform slightly under the influencethe overburden pressure. At the point where two gratouch, a small, tangential circle of contact is formed, tradius of which is given by the Hertz theory19 of elasticbodies in contact. Assuming that the two grains are identspheres of diameterug , Young’s modulusEg and Poisson’sratio ug , the radius,a, of the circle of contact is

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~12ug2!

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whereF is the force pressing the spheres together. At depdin the sediment, the forceF scales with the overburden presure, P, which is defined20 as the excess pressure arisifrom the difference between the bulk density of the sedimand the density of the pore water

P5~ro2rw!gd5~12N!~rg2rw!gd, ~9!

whereg is the acceleration due to gravity. The expressionthe right of Eq.~9! derives from Eq.~5! for the bulk density,which accounts for the dependence of the overburden psure on the porosity. It follows that the radius of the circlecontact may be expressed as

a}@~12N!dug#1/3, ~10!

where the constant of proportionality involves the elasproperties of the mineral grains but is independent ofbulk properties of the two-phase medium.

Now, the moduligp andgs scale with the rates at whicsliding events occur within the circle of contact of radiusa.With many micro-asperities distributed randomly over eaface of the contact area, the rate of sliding will scale withnumber of asperities available to slip against one anothethe case of the compressional modulus, most of the slidtakes place around the perimeter of the circle of contwhere the normal pressure is a minimum. Thus, the numof asperities involved in the sliding process is expectedscale with the perimeter of the circle of contact, and hegp is proportional toa. Shearing, on the other hand, involvea slip of the two flat faces of the surface of contact agaeach other, in which case the number of available asperscales with area of the circle of contact, and thereforegs is

J. Acoust. Soc. Am., Vol. 117, No. 1, January 2005

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proportional toa2. According to these arguments, the twmoduli may be expressed in terms of the porosity, grain sand depth in the sediment as follows:

gp5gpoF ~12N!ugd

~12No!ugodoG1/3

, ~11!

and

gs5gsoF ~12N!ugd

~12No!ugodoG2/3

, ~12!

where the terms in brackets have been constructed to bmensionless, with the factorsgpo and gso taking numericalvalues that are independent of the porosity, grain size,depth in the sediment. By introducing the three referenparameters, porosityNo , grain sizeugo , and depthdo , intothe denominators of Eqs.~11! and~12!, awkward dimensionsare avoided when the terms in square brackets are raisedfractional power. It should be clear that the values of thereference parameters may be chosen for convenience;do not represent additional unknowns. To distinguish thfrom the compressional and shear moduli, the paramegpo and gso , respectively, will be referred to as compresional and shear coefficients. Obviously,gpo andgso are thevalues ofgp andgs whenN5No , ug5ugo , andd5do .

In making the comparisons between the G-S theorydata, the values of the compressional and shear coefficieas well as the strain-hardening index,n, are held fixed for allsediments, from coarse sands to the finest silts and clayprinciple, however,gpo , gso , andn could depend weakly onthe elastic properties of the mineral grains and also on sfactors as the micro-roughness of the areas of contact. Tin practice, the values of these three parameters could dslightly from one sediment to another but, judging from tgood quality of the match between theory and data, as donstrated below, it would seem that such variations are vminor.

The functional dependence of the G-S dispersion retions on the macroscopic, geoacoustic parameters of a sment are given explicitly and completely by the algebraexpressions in Eqs.~1! to ~12!. From a casual inspection, it ievident that the selection of geoacoustic parameters apping in the G-S theory is quite different from that in Biottheory.13,14According to Table 1.1 in Stoll,21 the Biot theoryinvolves the permeability, the viscosity of the pore fluid, tpore-size parameter, the structure factor, the~complex! shearmodulus of the skeletal frame, and the~complex! bulk modu-lus of the skeletal frame, a total of eight parameters, allwhich are absent from the G-S formulation of the dispersrelations. Instead, the G-S theory includes the three~real!parameters representing microscopic processes occurringrain contacts,gpo , gso , andn. Through the compressionaand shear moduli@Eqs. ~11! and ~12!#, the G-S theory alsoincludes the grain size and the depth in the sediment, neiof which appears explicitly in Biot’s theory. The porosity, thbulk density, and the physical properties~densities and bulkmoduli! of the individual constituents of the biphasic mdium are common to both theories.

According to Eq.~5!, the bulk density and the porositare linearly related and hence are fully correlated. In fa

139Michael J. Buckingham: Wave properties of marine sediments

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Richardsonet al.10–12,22have used this relationship to compute the bulk density from laboratory measurements ofporosity. Hamilton,3,7,8 on the other hand, measured the de

FIG. 1. Compilation of density versus porosity data, mostly from HamiltThe red line is from Eq.~5! using the values ofrw andrg listed in Table II.~See Table I for the key to the symbols in this and all subsequent figur!


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sity directly from weight and volume, rather than computiit by substituting his independently measured porosity iEq. ~5!. Hamilton’s data may therefore be used as a tesEq. ~5!.

Figure 1 shows Hamilton’s published measurementsdensity and porosity for many types of marine sedimeranging from very fine silts and clays to coarse sands.~TableI is the key to all the symbols, representing published dcollected from the literature, which appear in Figs. 1 to 1!.Also shown in Fig. 1 is a plot of the linear relationship btween density and porosity in Eq.~5!, evaluated withrw

51005 kg/m3 andrg52730 kg/m3. Most of the data pointsfall on or very close to the line, indicating that the correlatirepresented by Eq.~5! holds accurately in practice, makingunnecessary to examine the functional dependence ofwave properties on both the porosity and the bulk densPorosity alone is considered below.

III. EVALUATION OF gpo , gso , AND n

As already discussed, in addition to the macroscovariables~porosity, grain size, and depth in the sediment!, theG-S dispersion relationships@Eqs.~1!–~4!# involve the three

.

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TABLE I. Key to the data symbols in Figs. 1–11.

Symbol Color Source

s Red Hamilton~Ref. 5! ~1972!, Table 1s Green Hamilton~Ref. 5! ~1972!, Table 2s Blue Hamilton~Ref. 5! ~1972!, Table 3s Magenta Hamilton~Ref. 3! ~1970!, Table 2s Cyan Hamilton~Ref. 8! ~1987!, Tables 1 & 2

* Blue Hamilton~Ref. 3! ~1970!, Table 1

* Cyan Hamilton~Ref. 8! ~1987!, Tables 3 & 41 Red Hamilton~Ref. 6! ~1980!, Table 11 Green Hamiltonet al. ~Ref. 2! ~1970!, Table 21 Blue Hamilton~Ref. 43! ~1963!, Table 1% Red Hamilton~Ref. 44! ~1956!, Tables 1a & 1b% Green Wood & Weston~Ref. 45! ~1964!,s Black Buckingham & Richardson~Ref. 23! ~2002!, Tables 1a &

1b1 Black Richardson~Ref. 12! ~2002!, Fig. 1 & Richardson~Ref. 33!

~2000! Fig. 2%—% Black Richardsonet al. ~Ref. 25! ~2001!, Fig. 15

*—* Black Richardson~Ref. 10! ~1991!, Table 1 & Fig. 31 Cyan Richardson~Ref. 46! ~1986!, Table 1

* Magenta Richardsonet al. ~Ref. 9! ~1991!, Tables 43.1 and 44.23 Red Richardson & Briggs~Ref. 22! ~1996!, Figs. 3, 5, 63 Blue Richardson~Ref. 11! ~1997!, Table 1, La Spezia & Adriatic

Sea3 Green Richardson~Ref. 11! ~1997!, Table 1, Panama City & Boca

Raton3 Magenta Richardson~Ref. 11! ~1997!, Table 1, Eckernfoerde Bay

Germany3 Cyan Richardson~Ref. 11! ~1997!, Table 1, Eel River & Orcas

Bay^ Green Brunson & Johnson~Ref. 47! ~1980!, Table II% Magenta Simpsonet al. ~Ref. 15! ~2003!, Fig. 6^ Blue Brunson~Ref. 48! ~1991!, Tables 1 & 2

* Black Richardson~Ref. 10! ~1991!, Table 1 & Fig. 31 Magenta Richardson & Briggs~Ref. 49! ~1993!, Table 13 Black McLeroy & DeLoach~Ref. 50! ~1968!, Table II

* Red McCann & McCann~Ref. 51! ~1969!, Table 3

* Green Richardson~Ref. 12! ~2002!, Fig. 3


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J. Acoust. Soc. Am.

TABLE II. Parameters appearing in the G-S dispersion relations.

Material parameter Symbol Value Comments

Density of pore fluid~kg/m3! r f 1005 Evaluated from density versusporosity fit to data, Fig. 1

Bulk modulus of pore fluid~Pa! K f 2.3743109 From Kaye & Laby~Ref. 52!, p. 35Density of grains~kg/m3! rs 2730 Evaluated from density porosity fit

to data, Fig. 1Bulk modulus of grains~Pa! Kt 3.631010 From Richardsonet al. ~Ref. 53!Compressional coefficient~Pa! gpo 3.8883108 Evaluated from spot-frequency

compressional and shear wave datShear coefficient~Pa! gso 4.5883107 Evaluated from spot-frequency

compressional and shear wave datStrain-hardening index n 0.0851 Evaluated from spot-frequency

compressional wave datarms grain roughness~mm! D 1 Evaluated from porosity versus

grain size distribution, Fig. 7~b!Reference grain diameter~mm! uo 1000 Arbitrary choiceReference depth in sediment~m! do 0.3 Arbitrary choiceReference porosity No 0.377 Arbitrary choicePorosity N VariableMean grain diameter~mm! ug VariableDepth in sediment~m! d Variable

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unknown constants,gpo , gso , andn, which characterize themicroscopic interactions occurring at grain boundaries ding the passage of compressional and shear wavespresent, no theory exists from which these constants coulcomputed, so instead they are evaluated here from sfrequency measurements of three wave properties: the cpressional speed, the compressional attenuation, andshear speed.

The two compressional-wave observations are tafrom Table 1a in Buckingham and Richardson23 for theSAX99 medium-sand [email protected] ~f51.27!,N50.377]: at depthd50.3 m and a frequency of 100 kHzthe measured sound speed~cf! and attenuation~af! are 1787m/s and 30.93 dB/m, respectively. The shear-speed meament, taken from Fig. 1 in Richardson,12 is for a well-sorted,fine-sand [email protected] ~f52.07!, N50.358] ata North Sea site designated C1: at depthd50.28 m and afrequency of 1 kHz, the~average! shear speed is 131.4 m/

When the measurements cited above are substitutedthe compressional and shear dispersion relations@Eqs. ~1!–~4!#, some elementary algebra yields the values ofgpo , gso ,andn listed in Table II. In the following comparisons of thG-S theory with numerous data sets, these values arefixed for all sediments. Under this constraint, it is of sominterest to examine how well the G-S dispersion relatioships represent the wave properties in sediments ranfrom coarse sands to very fine clays. This is achievedconsidering the dependencies of the wave speeds and atations on porosity and grain size. The depth dependencthe wave properties is also compared with data from varitypes of sediment. As will become evident, the fixed valuof gpo , gso , andn cited in Table II lead to theoretical waveproperty dependencies on porosity, grain size, and depthalign with essentially all the data. Before discussingphysical properties, however, the theoretical dependenciethe wave speeds and attenuations on frequency are compwith observations.

, Vol. 117, No. 1, January 2005

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As mentioned earlier, most of the data used in the flowing comparative discussions are fromin situ measure-ments made in silicilastic~quartz sand! sediments. Carbonatsediments are excluded, since they differ from silicilasmaterials in several ways,24 including having open, hollowgrains, which, with all else equal, gives rise to higher poroties. Hamilton and Bachman7 found that ‘‘hollow tests~shells! of Foraminifera act as solid particles in transmittinsound,’’ implying that the intraparticle porosity had littleany effect on the wave properties. A preliminary comparisbetween wave data from carbonate sediments and thetheory, evaluated using the interparticle~rather than total!porosity, appears to support Hamilton and Bachman’s cclusion but requires more detailed examination before bediscussed further.

IV. FREQUENCY DEPENDENCE

A. Compressional wave

Figure 2 showsin situ measurements of the frequencdependence, from 25 to 100 kHz, of the compressional wspeed and attenuation at the SAX99, medium-sand expment site off Fort Walton Beach in the northern GulfMexico.25 The smooth curves, representing the G-S theowere evaluated from Eqs.~1! and ~2!. These theoreticacurves pass precisely through the higher of each pair of dpoints at 100 kHz as expected, since both data points wused in evaluating the compressional and shear coefficigpo , gso as well as the strain-hardening index,n.

The sound-speed data from the SAX99 site@Fig. 2~a!#exhibit weak logarithmic dispersion at a level of approxmately 1% per decade of frequency. A similar logarithmgradient is predicted by the G-S theory. Although theresome scatter in the data~of the order of610 m/s!, it is clearthat the measured sound speeds cluster around the theorline. The observed SAX99 dispersion is consistent withcently reportedin situ measurements by Simpsonet al.,15


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which were made in a medium-sand sediment at St. AndrBay, Florida. Over the frequency band from 3 to 80 kHthey also observed logarithmic dispersion in the region1% per decade. Such a weak gradient is quite difficultdetect in situ, and earlier investigators not uncommonfailed to observe any variation of sound speed with fquency. Hamilton,4 for example, reported that ‘‘there is negligible dependence or no dependence of velocity onquency,’’ on the basis of which he developed his empirielastic model of wave propagation in sediments.

Visual inspection of the data in Fig. 2~b! reveals that,between 25 and 100 kHz, the compressional attenuation fthe SAX99 site scales almost exactly as the first powerfrequency. In fact, a straight line may be drawn throughdata points which, when extrapolated, passes precithrough the origin. However, it has long been known thapure linear scaling of attenuation with frequency is unphycal in that it violates causality.26–28 But, no such difficultyarises with the G-S theory, which yields an attenuation tha

FIG. 2. Measured frequency dependence of the compressional~a! speed~semilogarithmic axes! and ~b! attenuation~linear axes! for the medium-sand SAX99 site. The smooth, solid curves represent the predictions oG-S theory, computed from Eqs.~1! and~2! using the reported values~Ref.25! of the porosity, grain size, and [email protected], ug5414.7mm ~f51.27!, d50.3 m]. The dashed curves are from William’s EDF approximtion ~Ref. 29! to the Biot theory, withk52.5•10211 m2 andA51.35.


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almost but not quite linear in frequency, as exemplifiedFig. 2~b!. This near-linear curve, which rises marginalfaster than the first power of frequency, passes throughorigin and accurately through the data points.

The question of whether the attenuation in marine sements varies~nearly! linearly with frequency has been a controversial issue for many years. Hamilton8 consistently main-tained, on the basis of extensive measurements madehimself and others, that the attenuation exhibits a linear sing with frequency. Stoll,21 on the other hand, has long agued on the basis of the Biot theory13,14 that the attenuationdeviates significantly from a first-power dependence onquency, varying as the square root of frequency overbandwidth of the data shown in Fig. 2~b!.

Biot’s dispersion curves, evaluated from Williams’29

‘‘effective density fluid’’ ~EDF! approximation to Biot’s fulltheory, are included as the dashed lines in Fig. 2. To compthe EDF sound speed and attenuation curves, the valuetwo physical parameters peculiar to the Biot theory, the pmeability, k52.5310211m2 and tortuosityA51.35, weretaken from Table 1 in Williamset al.,30 which lists propertiesof the SAX99 sediment. In Fig. 2~a!, it can be seen that theEDF sound speed shows weak dispersion, aligning reaably with both the data and the G-S curve, although at lowfrequencies than those shown the two theoretical predictdiverge. The EDF attenuation in Fig. 2~b!, however, fails tomatch the data over most of the frequency range. Whethe attenuation data scale essentially linearly with frequenthe EDF attenuation curve exhibits the classic, higfrequencyf 1/2 dependence of a viscous fluid.

Since viscous dissipation of the pore fluid is the onloss mechanism in the Biot theory,13,14 it is not surprisingthat the frequency dependence of the Biot attenuation isactly the same as that of a true viscous fluid.31 Although thedetails of Biot’s attenuation curve may vary with materparameters such as permeability and tortuosity, the functioform is always the same: at low frequencies the attenuascales as the square of frequency, transitioning at abokHz ~depending on permeability and viscosity! to a square-root scaling with frequency. The discrepancy, illustratedFig. 2~b!, between such behavior and the near-linear scaof the attenuation data with frequency is a strong indicathat the viscous dissipation mechanism of the Biot themay be insignificant in marine sediments.

Evidence to support such a conclusion is found not oin the SAX99 attenuation data. From 3 to 80 kHz, Simpset al.15 have observed a~near! linear dependence of attenuation on frequency~their Fig. 6! in a medium-sand sedimenin St. Andrews Bay, Florida; and earlier laboratory expements in fine sand by Simpsonet al.32 returned a similar~near! linear scaling of the attenuation with frequency btween 4 and 100 kHz~their Fig. 7!. As illustrated in Fig.2~b!, Biot’s theory cannot account for a linear scalingattenuation with frequency; but, such behavior is consistwith Hamilton’s observations and also the predictions ofG-S theory.

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B. Shear wave

In situ measurements of shear-wave properties as futions of frequency are rarer than those of compressiowaves. In fact, no published,in situ measurements of sheawave dispersion and attenuation over extended frequeranges are known to the author.

The shear-wave speed and attenuation predicted fthe G-S dispersion relations in Eqs.~3! and~4! are shown inFig. 3. The curves were computed using values of the poity, grain size, and depth in the sediment appropriateRichardson’s12 North Sea site C1. The theoretical curve fthe shear speed in Fig. 3~a! shows dispersion at a level oapproximately 10% per decade of frequency, which is strger than the dispersion in the compressional wave by a faof 10 or so. It is evident from the uniform gradient of thcurve in Fig. 3~a! that the predicted shear dispersion is logrithmic as expected, sincecs scales asf n/2 in Eq. ~3!. Thelone shear-speed datum in Fig. 3~a! falls precisely on thetheoretical curve, since it was used in the evaluation ofcompressional and shear coefficients,gpo andgso . The pre-

FIG. 3. Shear-wave~a! speed and~b! attenuation versus frequency fromRichardson’s North Sea site C1, well-sorted fine sand. The solid lines wevaluated from Eqs.~3! and ~4! using the reported values~Ref. 33! of theporosity, grain size, and depth for the [email protected], ug5238.2mm ~f52.07!, d50.28 m].


c-al

cy

m

s-o

-or

-

e

dicted shear attenuation in Fig. 3~b! can be seen to exhibit a~near! linear dependence on frequency, varying asf 12n/2.Note that the theoretical curve passes below but close tosole reported data point33 taken at a frequency of 1 kHz. Thiagreement provides a mild test of the G-S theory, sinceshear attenuation was not used in computinggpo , gso , or n.A single data point, of course, provides no informationthe slope of the shear attenuation versus frequency cuwhich is unfortunate because the gradient~as in the case ofthe compressional attenuation! provides an important test otheoretical models of waves in sediments.In situ measure-ments of shear dispersion and attenuation over extendedquency ranges are challenging but would be extremely usfor future validation of theoretical predictions.

V. DEPTH DEPENDENCE


Publishedin situ data on the depth dependence of tsound speed and attenuation immediately beneathseawater–sediment interface are scarce. Hamilton34,35 dis-cussed the variation of the sound speed with depth, butcused mainly on significantly greater depths than the mor two of interest here. However, he did briefly consider tvariation of the sound speed in the first 20 m of sasediments,34 for which he developed the following empiricapower-law relationship, based on a curve fit to laboratodata:

cpH51806d0.015. ~13!

According to this expression, the sound speed is zero atseawater–sediment interface and increases rapidly in themeter or so, after which the rate of change slows signcantly. No allowance for any variation of the sound spewith porosity is made in Eq.~13!.

Hamilton36 also considered the attenuation of sound afunction of depth. Based on laboratory measurements37,38 ofthe variation of attenuation with confining pressure, he pposed an empirical inverse power-law relationship forattenuation versus depth in sand

apH50.45d21/6, ~14!

where the units ofapH are dB/m/kHz. Again, porosity isabsent from this empirical representation. According to E~14!, the attenuation is infinite at the seawater–sedimentterface but decreases very rapidly in the first few centimetIt seems from curve B in Hamilton’s36 Fig. 3 that he consid-ered Eq.~14! to be valid down to a depth of 150 m. Withithe first few meters of the interface, however, his data arepoorly resolved in depth to say whether they supportstrong negative gradient exhibited by Eq.~14!.

Since Hamilton’s8 final review of the acoustic propertieof sediments was published in 1987, fewin situ measure-ments of near-interface sound-speed and sound-attenuprofiles in sand sediments have appeared in the literatAcoustic profiles from cores, however, are available, somethe most recent being from Richardsonet al.25 for theSAX99 site. They measured sound speed and attenuatiofunctions of depth in a large number of cores at a freque

re


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of 400 kHz. The results are shown in Fig. 4, along with tG-S theory~smooth solid lines!, evaluated taking the porosity as independent of depth.39 Hamilton’s empirical expres-sions are also included~dashed lines! for comparison. Itshould be noted that the G-S attenuation was computedfrequency of 400 kHz and then divided by 400, to obtainattenuation profile shown in Fig. 4~b!, having units of dB/m/kHz. This procedure is consistent with that applied todata. For the purpose of making the comparison betwtheory and data in Fig. 4~b!, it would have been inappropriate to compute the attenuation directly for a frequency okHz, since this would have underestimated the theoretresult~relative to the data! due to the slight nonlinear dependence of the G-S attenuation on frequency@Fig. 2~b!#.

At each depth increment, the sound speed and attetion data show some spread, as indicated by the horizobars in Fig. 4. Note, however, that the variations in the sospeed are less than62%, whereas those in the attenuati

FIG. 4. Sound-speed~a! ratio and ~b! attenuation versus depth for thmedium-sand SAX99 site. The measurements were made on diver-collcores at 400 kHz and each horizontal line represents the spread of theat a given depth. The smooth solid lines, representing the G-S theory,evaluated from Eqs.~1! and ~2! using the reported values~Ref. 25! of theporosity, grain size, and measurement [email protected], ug

5414.7mm ~f51.27!, f 5400 kHz]. The dashed lines, depicting Hamton’s empirical profiles, were evaluated from Eqs.~13! and ~14!.


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are of the order of640%. As discussed below, this relativehigh variability in the attenuation data may be attributablevarious factors including scattering from inclusions suchshell fragments in the sediment as well as random couplosses, which are expected whenever a sensor is inserteda granular sedimentary medium.

Below a depth of 10 cm, the G-S theory yields a sounspeed profile that falls within the limits of the data in Fi4~a!. At shallower depths, where the gradient in the souspeed is very high, the G-S theory falls a little below tlower limits of the data. This small discrepancy betweentheory and the data may be due to the difficulty of measurthe sound speed near the interface, where the gradient oupward-refracting profile is particularly steep: diving wavmay reach the receiver first, giving rise to an overestimatethe sound speed. Note that the trends of the G-S sound-sprofile and Hamilton’s empirical relationship are similawith both showing a rapid increase within a few centimetof the interface. In the case of the G-S profile, this hignear-interface gradient is an effect of the overburden psure, which gives rise to a high rate of increase of the copressional coefficient@Eq. ~11!# immediately below theboundary. At the boundary itself (d50), the G-S profiletakes Wood’s value,co , whereas Hamilton’s expression goeto zero.

Turning to the attenuation profiles in Fig. 4~b!, it appearsthat the negative gradient of Hamilton’s empirical curvenot consistent with the data~even allowing for the fact thathe data show considerable spread!. On the other hand, theG-S attenuation profile accurately tracks the lower-limitivalues of the measured attenuation. As will become apparthe G-S theory systematically delineates the lower limitsall the ~compressional and shear! attenuation data sets examined in this paper, that is, the G-S theory traces the lowbound to the envelope of the measured attenuation valu

A plausible interpretation of this observation is that tG-S theory predicts theintrinsic attenuation due to intergranular interactions, which convert acoustic energy iheat, whereas the measurements represent theeffectiveat-tenuation, that is, the total attenuation experienced bsound or shear wave in propagating through the granmaterial. Thus, in addition to the intrinsic attenuation, teffective attenuation includes all other losses, notably thdue to scattering from inhomogeneities such as shell frments in the medium. Naturally, since shells and other sterers are likely to be distributed randomly throughout tsediment, the effective attenuation is expected to be higvariable, but with a lower limiting value equal to the intrinsattenuation. According to this argument, the G-S theoryexpected to coincide with the lower bound of the data,indeed it does in Fig. 4~b!.

B. Shear wave

Figure 5~a! shows Richardson’s12 in situ measurementsof the shear speed versus depth at 1 kHz in well-sortedsand at the North Sea C1 site. At each depth, the data poiFig. 5~a! represents the mean of the several measured vareported by Richardson.12 The data display a distinct positivgradient, with the shear speed approximately doubling

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tween depths of 5 and 30 cm. Similar behavior is predicby the G-S theory@Eqs. ~3! and ~12!#, as shown by thesmooth solid line in Fig. 5~a!, which increases as the cubroot of depth in the sediment. In evaluating the G-S lineFig. 5~a!, the porosity was taken to be independent of depconsistent with the porosity profiles measured on cores25 andin situ39 during the SAX99 experiment. It is clear that ththeoretical curve in Fig. 5~a! closely matches the experimentally determined shear-speed profile.

The cube root of depth power law from the G-S theorysimilar in form to an empirical shear-speed profile for sasediments proposed by Hamilton8

csH5128d0.28. ~15!

This expression is plotted in Fig. 5~a! as the dashed line. Infact, Hamilton8 discussed several empirical expressions hing the form of Eq.~15! but with slightly different exponentsranging from 0.25 to 0.312. Over the limited depth rangeRichardson’s data,12 these exponents are almost indisti

FIG. 5. Shear-wave~a! speed and~b! attenuation versus depth for the welsorted fine sand at North Sea site C1. The smooth solid lines, represethe G-S theory, were evaluated from Eqs.~3! and ~4! along with Eq.~12!using the reported values~Ref. 33! of the porosity, grain size, and measurment [email protected], ug5238.2mm ~f52.07!, f 51 kHz]. Thedashed lines are Hamilton’s empirical profiles, as given by Eqs.~15! and~16!.


d

,

d

-

f

guishable, not only from one another but also from the Gexponent of 1/3. However, Hamilton’s~fixed! coefficient of128 in Eq. ~15! significantly underestimates Richardsondata in Fig. 5~a!, by approximately 30% at a depth of 30 cmIt should be noted that, according to Eq.~12!, the corre-sponding coefficient in the G-S theory is a function of prosity and grain size, and thus may differ from one sedimto another. Equation~15! obviously does not accommodasuch behavior.

The shear attenuation measured by Richardson33 at theNorth Sea C1 site is shown in Fig. 5~b!. Included in thefigure is the G-S theoretical prediction evaluated from E~4! and ~12!, again taking the porosity as independentdepth.25,39 It is evident from these equations that the prdicted attenuation scales as the reciprocal of the cube roodepth and that the coefficient of proportionality is a functiof porosity and grain size. Below a depth of 10 cm, the Gtheory follows the lower bound of the attenuation dawhich is consistent with the earlier argument that the Gcurve depicts intrinsic attenuation, as opposed to the eftive attenuation represented by the data. At shallower depthe G-S theory overestimates the measured attenuationRichardson33 has suggested that these near-interface dpoints may in fact be too low as a result of ducting~divingwaves! produced by the steep shear-speed gradient withinupper few centimeters of the sediment.

Hamilton8 has proposed the following empirical expresion for the shear attenuation profile:

asH524d21/6, ~16!

which is shown as the dashed line in Fig. 5~b!. Note thatHamilton’s inverse-fractional-power-law scalings for thcompressional attenuation@Eq. ~14!# and shear attenuatio@Eq. ~16!# are exactly the same, both having an exponen21/6. Like the G-S shear attenuation profile, Eq.~16! showsa steep negative gradient immediately below the interfawhereas the gradient of Richardson’s data is weakly positAs mentioned above, this discrepancy may be the resulducting just below the interface.

Figure 6 shows the shear-speed profiles of three fingrained sediments, measuredin situ by Richardsonet al.10 atBoa Dragaggio~fine sands and silt clays!, Viareggio ~siltclay!, and Portovenere~silt and clay!. ~No shear-attenuationdata were reported for these sites.! In all three cases, theshear speed is significantly less than that at the fine-sNorth Sea C1 site shown in Fig. 5~a!. Notice the subtle dif-ferences between the Boa Dragaggio, Viareggio, and Ptovenere profiles, and the fact that all three profiles are acrately reproduced by the G-S theory, evaluated from Eqs.~3!and ~12! using the reported porosities and grain sizes.

The difference between the observed shear-speedfiles at Boa Dragaggio and Vareggio@Figs. 6~a! and 6~b!# isparticularly interesting, since the reported porosities of thtwo sediments are almost the same, at 0.57 and 0.58, restively. It follows that the difference in the shear speeds attwo sites, amounting to about 10 m/s at a depth of 2 m, mbe due to the difference between the mean grain sizes.Dragaggio is the coarser of the two sediments, with aported mean grain diameter10 of 3.4 mm ~f58.2!, compared

ing


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FIG. 6. Shear-speed profiles from~a! Boa Draggagio @N50.57, ug

53.4 micron ~f58.2!, f 51 kHz]; ~b! Viareggio @N50.58, ug

51.82 micron ~f59.1!, f 51 kHz]; and ~c! [email protected], ug

51.05 micron~f59.9!, f 5300 Hz]. The smooth curves, representing tG-S theory, were evaluated from Eqs.~3! along with Eq.~12! using thereported values~Ref. 10! of the porosity, grain size, and measurement fquency for each of the sites, as listed here in square brackets.


with 1.82 mm ~f59.1! at Viareggio. In the G-S theory, thgrain size affects the shear speed only through the smodulus,gs , given by Eq.~12!. According to this expres-sion, the shear modulus for Boa Dragaggio is greater tthat for Viareggio by a factor of 1.5. This is sufficient to givrise to slightly dissimilar theoretical shear-speed profilesthe two sites, in excellent agreement with the data, as shin Figs. 6~a! and 6~b!.

At the Portovenere site@Fig. 6~c!#, the porosity is mar-ginally higher and the grain size a little lower than at BDragaggio and Viareggio. The enhanced porosity affectsbulk density,ro in Eq. ~5!, and also the shear modulus,gs inEq. ~12!, while the smaller grain size influences onlygs . Thenet effect, as predicted by Eq.~3!, is a slightly slower shear-speed profile than that at either of the other two sites. Agit can be seen that the theoretical curve and the data arvery good agreement.

VI. RELATIONSHIPS BETWEEN PHYSICALPROPERTIES

Although the porosity, grain size, and bulk density apear independently in the G-S dispersion expressions, thquantities are not themselves independent: finer-grain sments tend to have lower densities and higher porositiescoarser materials. In order to evaluate the G-S dispersrelations as a function of any one of the three physical prerties, porosity, grain size, and density, it is necessary firsestablish the relationships that exist between them.

Of these inter-relationships, that between bulk densand porosity is well known and has already been introduas the linear combination of the two constituent densitiesEq. ~5!. This expression accurately matches the data, asemplified in Fig. 1. It follows that, provided the densitiesthe fluid and mineral phases are known, the bulk densitybe determined directly from measurements of the poroand vice versa.

The functional dependence of porosity on grain sizemore difficult to treat, not least because porosity is nuniquely determined by grain size: sediments with identiporosities may exhibit mean grain sizes that differ from oanother, as exemplified by the sediments investigatedRichardsonet al.10 at Boa Draggagio and Viareggio@seeFigs. 6~a! and~b!#. Interestingly, Hamilton’s data on porositversus grain size show a significantly smaller scatter thancorresponding data collected from numerous sitesRichardson, a difference that becomes clear on compaFigs. 7~a! and 7~b!. However, a trend common to both dasets is that finer-grain sediments tend to exhibit higherrosities.

Any variation of the porosity with grain size representsdeparture from the way smooth, uniform spheres packgether. Hamilton8 has attributed the observed variationporosity with grain size, illustrated in Fig. 7, to several fators including nonuniformity in the size and the shape ofgrains.

Amongst unconsolidated marine sediments, the lowporosities are found in the coarse sands, almost alwaysing values close to 0.37. As it happens, 0.37 is also therosity of a random ‘‘close’’ packing of smooth, uniform


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spheres,40,41 which suggests that, in the coarser sedimegrain shape~or roughness! effects represent a negligible departure from sphericity, and thus the packing is much likrandom packing of smooth spheres. In the finer-grained sments, on the other hand, surface roughness may be corable with or much greater than the mean grain diametewhich case close contact between adjacent grains isvented, thus allowing pore water to percolate between grawhich results in an increase in the porosity. Grain ‘‘shapand ‘‘roughness’’ in this context cover a multitude of nospherical conditions, encompassing smooth, potato-grains, high-aspect-ratio platelets, and very spiky, hedgehlike particles.

Based on these ideas~i.e., a random packing of rougspheres!, Buckingham42 developed the following relationshibetween porosity and grain size:

N512PsH ug12D

ug14DJ 3

, ~17!

FIG. 7. Porosity versus grain size.~a! Comparison of Hamilton’s data withEq. ~17!, evaluated takingD53 mm ~red curve!. ~b! Comparison of Rich-ardson’s data with Eq.~17!, takingD51 mm ~red curve!.


s,

ai-

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eg-

wherePs50.63 is the packing factor of a random arrangment of smooth spheres andD is the rms roughness measured about the mean~equivalent volume sphere! surface ofthe grains. The inverse of Eq.~17! gives the grain size asfunction of the porosity

ug52D~2B21!

12B, ~18!

where

B5H 12N

12NminJ 1/3

. ~19!

According to Eq.~17!, when the grain size is very mucgreater than the rms roughness, the porosity approacheminimum value, Nmin512Ps50.37, appropriate to thecoarse-sand regime of Fig. 7. At the other extreme, whenroughness is very much greater than the grain diametewith the high-aspect-ratio clay and silt particles, the porostakes its maximum value,Nmax512(Ps/8)50.92, whichconforms with the fine-grain, high-porosity data in Fig. 7.

It is clear from the presence ofD, the rms roughnessparameter, in Eq.~17! that sediments of nominally the sammean grain size may exhibit different porosities. A shasand having very rough grains~high D! may be considerablymore porous than an otherwise similar smooth-grained s~low D!. Although the rms roughness of the grains is noparameter that is normally reported, the value ofD53 mmyields a relationship between porosity and grain size frEq. ~17! that follows the average trend of Hamilton’s dataFig. 7~a!, whereas a somewhat lower value ofD51 mm ismore appropriate to the average of Richardson’s data, alustrated in Fig. 7~b!.

It is not clear why Hamilton’s and Richardson’s data sin Figs. 7~a! and 7~b!, respectively, should differ so markedly. The differences appear not only in the average trebut also in the scatter of the data about the mean poroespecially for the finer-grained materials, a spread whichnoticeably greater in Richardson’s data. Perhaps the reafor the disparities between the two data sets is nothing mthan coincidence in that the majority of the sediments exained by Richardson just happened to have smoother graand hence lower porosities, than those analyzed by Haton.

In order to make comparisons between the theoretand measured wave properties as functions of the poroand the grain size, an ‘‘optimum’’ value ofD must be se-lected for substitution into Eq.~17!. Since most of the data inthe following comparisons stem from Richardson’s measuments, the value ofD51 mm is adopted for the rms roughness parameter, consistent with the comparison betweenand theory in Fig. 7~b!. With a fixed value ofD in Eq. ~17!,the theoretical predictions of the wave properties fromG-S dispersion relations are, of course, single valued inporosity and in the grain size, but the resultant curvesuseful for comparison with the average trends of the muvalued experimental data.


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VII. POROSITY DEPENDENCE


The ratio of the sound speed in the sediment dividedthat in the water column is much less sensitive to tempeture variations amongst sediments than the sound speeself. Figure 8~a! shows a plot of the sound-speed ratio versporosity computed from the G-S theory@Eq. ~1!#, along withdata from a large number of sediments. The abundancdata points in Fig. 8~a! reveals a clear trend over the porosrange fromN'0.37 to N'0.92, covering coarse sandsclays. Throughout this range, the theoretical curve accurafollows the trend of the data.

An interesting feature of both the data and the theorcal curve in Fig. 8~a! is the extremely steep gradient of thsound speed at porosities corresponding to the coarser s(0.37,N,0.4). As the porosity rises marginally abovelower limit of 0.37, the sound speed plummets, eventuapassing through a broad minimum aroundN'0.8. In view ofthe sensitivity to the porosity in the coarser materials, itessential to have a high-precision measure ofN available if

FIG. 8. ~a! Sound-speed ratio versus porosity and~b! sound attenuationversus porosity. The red curves were evaluated from the G-S theory@Eq. ~1!#with f 538 kHz andd50.3 m, andD51 mm in Eq. ~17!. To obtain theattenuation in dB/m/kHz, for comparison with the data, the value compuat 38 kHz was divided by 38.


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the sound speed for a sand sediment is to be predicted arately from the G-S theory.

The compressional attenuation as a function of porois shown in Fig. 8~b!, where the smooth curve was computby evaluating the G-S attenuation@Eq. ~2!# for a frequency of38 kHz ~i.e., the frequency at which most of the data wecollected! and dividing the result by 38 to obtain the prdicted attenuation in dB/m/kHz. It is evident that the scatis much higher in the attenuation data@Fig. 8~b!# than in thesound-speed data@Fig. 8~a!#. As with the depth profiles ofattenuation in Fig. 4~b! ~compressional! and Fig. 5~b!~shear!, the G-S theoretical curve tracks the lower boundthe widely spread attenuation data in Fig. 8~b! ~if allowanceis made for relatively large measurement errors in the lowattenuations!. Such behavior is consistent with the fact ththe theory predicts the intrinsic attenuation, due to the irversible conversion of wave energy into heat, whereasmeasurements represent the randomly distributed effecattenuation. As discussed earlier, in addition to the intrinattenuation, the effective attenuation includes all other ty

FIG. 9. ~a! Shear-wave speed versus porosity and~b! shear-wave attenuationversus porosity. The red curves were evaluated from the G-S theory@Eqs.~3! and~4!# with f 51 kHz andd50.3 m, andD51 mm in Eq. ~17!. @Note:The shear wave data reported in several of Hamilton’s papers were mdetermined indirectly from measurements of interface waves. As theypear to be distinctly different in character from more recent direct measments of shear wave properties, his data are not included in these plo#

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of loss, due for instance to scattering from inhomogeneisuch as shell fragments in the medium.

Like the sound speed, the theoretical intrinsic attention decays extremely rapidly as the porosity rises incremtally above the minimum value of 0.37, with a similar rapdecay exhibited by the data. It follows that, in order to pdict accurately the intrinsic attenuation in a sand sedimenhigh-precision estimate of the porosity is required.

B. Shear wave

Figure 9~a! shows the speed of the shear wave as a fution of porosity, with the smooth curve representing the Gtheory @Eq. ~3!#. The clear downward trend in the datareproduced well by the theoretical curve. The finer, higporosity sediments exhibit the slowest shear speeds,values as low as 10 m/s in the highest porosity materials,clays. At the opposite extreme, the coarse sands with poties around 0.4 show shear speeds in excess of 100 m/with the compressional speed and attenuation, the shwave speed decays extremely rapidly in the coarse mateas the porosity increases slightly above its lowest value0.37. This behavior can be clearly seen in both the theorecurve and the data. Because of the very high gradienporosities in the vicinity of 0.4, a high-precision measument of porosity would be required in order to predict accrately the speed of the shear wave in a sand sediment.

Figure 9~b! shows the shear attenuation as a functionporosity, with the smooth curve representing the predictof the G-S theory@Eq. ~4!#. Relatively few data points appeain Fig. 9~b!, reflecting the difficulty of makingin situ shear-attenuation measurements. Nevertheless, sufficient datapresent to identify a lower bound to the effective attenuatia boundary which is accurately traced by the intrinsic atteation curve from the G-S theory. Apart from one errant poatN50.63, the data lie on or above the theoretical line, agconsistent with the idea that scattering and other loss menisms may add to the intrinsic attenuation predicted bytheory to yield the effective attenuation of the measureme

VIII. GRAIN-SIZE DEPENDENCE


Figure 10~a! shows the sound-speed ratio as a functof grain size, with the smooth curve representing the Gtheory@Eq. ~1!#. The data are well distributed throughout thfull range of grain sizes, from clays to coarse sands,show a distinct upward trend with increasing grain size. TG-S theory follows the average trend of the data very safactorily.

The compressional attenuation as a function of the gsize is shown in Fig. 10~b!, where the smooth curve represents the G-S theory@Eq. ~2!#, which was evaluated for afrequency of 38 kHz~i.e., the measurement frequency fmuch of the data! and divided by 38 to obtain dB/m/kHzThroughout the range of grain sizes, from clays to coasands, the data points show a high degree of scatter aspected, since the data represent the effective attenuation


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lower boundary of the envelope occupied by the data iscurately traced by the theoretical curve computed fromG-S theory for the intrinsic attenuation.

B. Shear wave

Data on shear-wave properties as a function of grain sare less abundant than those on the compressional waveshear speed versus mean grain diameter is plotted in11~a!, where the smooth curve represents the G-S theory@Eq.~3!# evaluated at a frequency of 1 kHz. Although a gap apears in the distribution of the data points between 15 andmm, a strong upward trend in the measured shear speedincreasing grain size is still easy to distinguish. Similar bhavior is exhibited by the G-S theoretical curve.

FIG. 10. ~a! Sound-speed ratio versus mean grain diameter and~b! soundattenuation versus mean grain diameter. The red curves were evaluatedthe G-S theory@Eqs.~1! and ~2!# with f 538 kHz andd50.3 m, andD51mm in Eq.~17!. To obtain the attenuation in dB/m/kHz, for comparison withe data, the value computed at 38 kHz was divided by 38.@Note: Hamil-ton’s data have been included in these plots because the G-S theory anexperimental data indicate an insensitivity to the grain roughness paramD that is, Hamilton’s and Richardson’s data follow much the same tren#


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As pointed out by Richardson,12 shear attenuation habeen measured at far fewer sites than shear speed. Tmeasurements of shear attenuation that are available areted in Fig. 11~b! against the mean grain diameter. Tsmooth curve in the diagram represents the G-S theevaluated for a frequency of 1 kHz. Many of the data poifall close to or above the theoretical curve, again consiswith the idea that the data represent effective attenuationlower limiting value of which is the intrinsic attenuatioyielded by the theory.

However, a group of data points in Fig. 11~b!, represent-ing very-fine-grained materials, silts and clays, with megrain diameters in the range between 1 and 10mm, fallsnoticeably below the theoretical curve. It is not clear wthese particular data points are seemingly too low~or thetheoretical curve too high!, especially as the very same mesured values of shear attenuation lie above the theorecurve in the plot of shear attenuation versus porosity in F

FIG. 11. ~a! Shear-wave speed versus mean grain diameter and~b! shear-wave attenuation versus mean grain diameter. The red curves were evafrom the G-S theory@Eqs. ~3! and ~4!# with f 51 kHz andd50.3 m, andD51 mm in Eq.~17!. @Note: The comment on Hamilton’s shear wave datathe legend to Fig. 9 also applies here.#


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9~b!. The data points in question in Fig. 9~b! fall in the po-rosity range from approximately 0.55 to 0.6. In view of thextreme difficulty of making precisionin situ measurementsof shear attenuation in very-fine-grained sediments~shearspeeds less than 50 m/s!, the explanation for the apparentllow data points in Fig. 11~b! may simply be a high level ofuncertainty in these measurements. Since no error barsreported with the original data set, it is difficult to ascertawhether this interpretation is plausible but the anomshould be resolved, one way or the other, as morein situshear-attenuation data become available.

IX. A SHEAR-WAVE INVARIANT

If the product of the G-S expressions for the shear sp@Eq. ~3!# and intrinsic attenuation@Eq. ~4!# is formed, theresultant expression is

csas52p f tanS np

4 D50.42f , ~20!

wheref is frequency~Hz! and the value ofn listed in Table IIhas been used to evaluate the scaling constant~0.42! on theright. According to Eq.~20!, the product of the shear speeand attenuation is directly proportional to the frequency. Tscaling constant is a function only of the strain-hardencoefficientn, being independent of all the macroscopic mterial properties of the sediment, that is, the porosity,grain size, the density, and the overburden pressure. Sinnrepresents microscopic processes occurring at gcontacts,17 including strain hardening in the molecularly thlayer of fluid separating asperities, it is expected to be esstially constant for all sediments composed of similar mateals, for instance, quartz grains and seawater. Thus, thestant 0.42 in Eq.~20! should hold for all silicilastic marinesediments

In principle, the predicted invariance of the shear spetimes the shear attenuation provides a good test of thetheory. In practice, however, the number of sediment siteswhich both shear-wave properties are available is almvanishingly small, making such a test impracticablepresent.

X. CONCLUDING REMARKS

The properties of the phase speed and attenuationcompressional and shear waves in marine sedimentsbeen examined in this article. Detailed comparisons hbeen made between measured wave properties, takenthe literature, and the predictions of a recently developgrain-shearing~G-S! theory of wave propagation in saturateporous media. The theory takes the form of two disperspairs, that is, four expressions, representing the phase sand attenuation of the compressional wave and the swave. In addition to frequency, the four G-S dispersion retionships are functions of porosity, density, and grain siand also overburden pressure, which translates into depthe sediment.

Besides the material properties, the theoretical expsions involve three real parameters, representing microscshearing processes that are postulated to occur at grain

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tacts during the passage of a wave. Two of these paramethe compressional and shear coefficients, are analogouthe Lamecoefficients of elasticity theory, and the third isnumerical index representing the phenomenon of strain hening in the molecularly thin layer of pore fluid sandwichbetween grain contacts. Each of the three grain-shearingrameters has been assigned a value, which was held fixeall the comparisons with data. Thus, the only available vables in the theoretical dispersion relationships are the mrial properties~porosity, grain size, and overburden pressu!and the frequency.

The compressional and shear-wave speeds and attetions have been plotted as functions of frequency, depththe sediment, porosity, and grain size, in each case withother variables held constant. In every comparison,theory accurately matches the average trend of the data sis to be emphasized that no adjustable parameters were aable in the theory to help achieve these multiple fits to da

The high quality of the match to data even holds for tattenuation of both the compressional and the shear wthe theory reliably traces out thelower boundto the highlyvariable attenuation values returned by measurementstraightforward interpretation of this observation is thattheory predicts theintrinsic attenuation, which arises fromthe irreversible conversion of wave energy into heat, wherthe measurements yield theeffectiveattenuation, that is, theintrinsic attenuation plus any additional sources of loss, sas scattering from inhomogeneities in the granular mediSince inclusions such as shell fragments tend to be randodistributed in the sediment, the effective~i.e., measured! at-tenuation shows large fluctuations, taking values thatbounded from below by the intrinsic attenuation, whichstable and well predicted by the theory. In the plots ofcompressional attenuation as a function of depth, forample, the theory accurately traces out the profile of the loest attenuation values observed at each depth@Fig. 4~b!#.

It is concluded that the G-S dispersion relationshipsEqs.~1! to ~4!, with the three fixed parametersgpo , gso , andn taking the values listed in Table I, are useful as a practtool for making estimates of the wave and physical propties of a sediment~excluding the effective attenuation, ocourse, but including the intrinsic attenuation!. For instance,if the sound speed were known, the shear speed, both~intrin-sic! attenuations, the porosity, and the density couldevaluated immediately and uniquely from the G-S theoretexpressions. Similarly, if the porosity were available frosay, a core sample, the wave speeds and~intrinsic! attenua-tions could be evaluated from the G-S dispersion relatioSuch utility has potential as the basis of a variety of invsion techniques for estimating the geoacoustic parametemarine sediments.

ACKNOWLEDGMENTS

Throughout the evolution of this work, Dr. Michael Rchardson, NRL Stennis, has been generous with his sedimdata as well as providing valuable guidance on the mateproperties of sediments. The research was supported byEllen Livingston, Ocean Acoustics Code, the Office of NaResearch, under Grant Number N00014-04-1-0063.


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