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CorrelationsbetweenPoisson'sratioandseismicwavevelocitiesforsomecommonrocksandsulfideores

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Tectonophysics 469 (2009) 61–72

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Correlations between compressional and shear wave velocities and correspondingPoisson's ratios for some common rocks and sulfide ores

Qian Wang, Shaocheng Ji ⁎, Shengsi Sun, Denis MarcotteDépartement des Génies Civil, Géologique et des Mines, École Polytechnique de Montréal, Montréal, Québec, Canada H3C 3A7

⁎ Corresponding author.E-mail address: sji@polymtl.ca (S. Ji).

0040-1951/$ – see front matter © 2009 Elsevier B.V. Adoi:10.1016/j.tecto.2009.01.025

a b s t r a c t

a r t i c l e i n f o

Article history:Received 15 August 2008Received in revised form 19 January 2009Accepted 27 January 2009Available online 7 February 2009

Keywords:Seismic velocitiesPoisson's ratioShear modulusYoung's modulusRocksSulfides

The correlations between Vp and Vs and corresponding Poisson's ratios are important in modeling andinterpreting seismic data in terms of chemistry and lithology. Data from high-pressure laboratorymeasurementsof compressional and shear wave velocities (Vp and Vs) are analyzed for 12 common categories of rocks(amphibolite, anorthosite, basalt, diorite, eclogite, felsic gneiss, gabbro-diabase, granite, intermediate gneiss,limestone, mafic gneiss, and peridotite) and 4 types of massive sulfide ores (chalcopyrite, pyrite, sphalerite andpyrrhotite). The analysis reveals that the linear correlation provides gooddescriptions for theVs–Vp and lnVs–lnVprelationships. Poisson's ratio is linearly correlated with Vs, Vp, shear modulus (G) and Young's modulus (E) forthese rocks and sulfide ores. A decrease in Poisson's ratio is associatedwith increases in Vs, G and E. However, thevariation of Poisson's ratio with Vp depends on the logarithmic ratio Rs/p(i.e., ∂lnVs/∂lnVp). Poisson's ratioincreases or decreases with Vp when Rs/pb1 or N1. Rs/p varies systematically with lithology (0.300 for granite,0.573 for diorite, 0.602 for felsic gneiss, 0.631 for intermediate gneiss, 0.721 for gabbro-diabase, 0.768 for maficgneiss, 0.866 for eclogite, 0.890 for amphibolite, and 1.391 for peridotite). It is suggested that Rs/p can be used as aproxy for the composition of the deep continental crust and the upper mantle.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

Seismic properties of rocks provide important constraints for thechemical composition, physical state and structure of the Earth'sinterior, especially for depths greater than 10–12 km (i.e., depthsbelow which rock samples have not yet obtained by drilling).Successful interpretation of seismic data in terms of radial and lateralvariations of chemical composition, mineralogy, temperature andpressure requires statistical information on the relationships betweenP- and S-wave velocities (Vp and Vs) and corresponding Poisson's ratioof each main category of lithology under the elevated conditions thatcharacterize the Earth's interior (e.g., Zandt and Ammon, 1995;Christensen, 1996; Chevrot and Van der Hilst, 2000; Ji et al., 2009).

For a statistically isotropic (i.e., texture-free polycrystalline rocks),linearly elastic solid, only two of the following elastic constants arerequired to characterize the seismic properties: the Young's (E), shear(G) and bulk (K) moduli and Poisson's ratio (υ). Once two independentconstants are known, other constants can be calculated according to thewell-known equations in elastic mechanics. The magnitudes of theelastic constants reflect crystal interatomic bonding, grain boundarystrength, andmicrostructural characteristics (e.g., microcracks, porosity,phase connectivity and continuity) of rocks during uniaxial extension/compression, simple shear and hydrostatic compression. If any new

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relationship is found to exist between these elastic constants for a givenisotropicmaterial, itwould bepossible to reduce thenumber of requiredindependent elastic constants from 2 to 1. This subject has receivedmuch attention inmaterials science. For example, by plottingG against Efor 34 metals including body-centred cubic (bcc), face-centred cubic(fcc) and hexagonal close-packed (hcp) crystal structures, Ledbetter(1977) obtainedG/E≈3/8. Gorecki (1980) carried out a statistical studyon the relationships between E, G, and K for polycrystalline metals withdifferent structures (Table 1). Kumar et al. (2003) reported a negativecorrelation between Poisson's ratio and ultrasonic shear wave velocityfor metals, alloys, intermetallics, ceramics, glasses, and polymers.Whether does such a correlation exist for natural polymineralic rocksthat constitute the Earth's crust and uppermantle? The first objective ofthe present study is to seek some hidden relationship between the twoindependent elastic constants for common categories of crustal andupper mantle rocks.

The elastic constants of a rock are usually determined throughacoustic measurements of P- and S-wave velocities (Vp and Vs) usingthe pulse transmission technique (Birch, 1960; Christensen, 1974;Kern, 1982; Burlini and Fountain, 1993; Ji and Salisbury, 1993;Watanabe et al., 2007), along with the density (ρ). In the literature,however, there are muchmore laboratory measurements of Vp than Vs

for rock samples (see Handbook of Seismic Properties of Minerals,Rocks and Ores, Ji et al., 2002). This is in part becausemeasurements ofVs met appreciable difficulties at low pressures, as the transduction ofshear wave through the specimen requires a firm contact between the

Table 1Statistical results of the relationships between E, G, and K for polycrystalline metalswith different structures.

Structure G/E G/K E/K

Face-centred cubic (fcc) 0.385 0.379 0.944Body-centred cubic (bcc) 0.357 0.373 1.041Hexagonal close-packed (hcp) 0.389 0.527 1.314

Original data from Gorecki (1980).

Table 2Parameters of the linear relations for best-fitting the Vs–Vp and lnVs–lnVp data.

Lithology P (MPa) N Density Vs=aVp+b lnVs=alnVp+b

g/cm3 a b R a b R

RocksAmphibolite 600 29 3.01 0.501 0.451 0.718 0.890 −0.356 0.724Anorthosite 600 8 2.78 0.632 −0.678 0.847 1.159 −0.936 0.841Basalt 600 21 2.89 0.622 −0.382 0.916 1.096 −0.755 0.916Diorite 600 9 2.76 0.337 1.555 0.820 0.573 0.248 0.822Eclogite 600 55 3.47 0.496 0.580 0.789 0.866 −0.286 0.782Felsic gneiss 600 67 2.70 0.346 1.462 0.593 0.602 0.183 0.592Gabbro-diabase

600 47 2.98 0.401 1.068 0.770 0.721 −0.047 0.771

Granite 600 31 2.66 0.172 2.575 0.215 0.300 0.744 0.218Intermediategneiss

600 14 2.76 0.366 1.371 0.419 0.631 0.140 0.419

Limestone 600 26 2.72 0.482 0.217 0.645 0.916 −0.506 0.647Mafic gneiss 600 70 3.06 0.426 0.924 0.751 0.768 −0.132 0.756Peridotite 600 65 3.09 0.728 −1.380 0.976 1.391 −1.400 0.981

Sulfide oresChalcopyrite 600 11 3.57 1.032 −2.764 0.913 1.846 −2.080 0.924Pyrite 300 12 4.18 0.908 −2.111 0.968 1.524 −1.524 0.968Pyrrhotite 600 24 4.16 0.403 0.965 0.966 0.695 −0.027 0.964Sphalerite 600 9 3.76 0.704 −0.872 0.932 1.280 −1.086 0.924

62 Q. Wang et al. / Tectonophysics 469 (2009) 61–72

transducer and the end surfaces of the specimen. In-situ seismicrefraction experiments often report only Vp data because theexperiments commonly use short period vertical seismometers fromwhich it is difficult to pick the S arrivals (e.g., Chevrot and Van Hilst,2000). Vp and Vs structures of the Earth's interior for a given regionare usually determined separated with typical frequency of ~1 s for Pwaves and ~20 s or longer for S waves. Moreover, raypaths used toconstrain these velocity structures may not be uniform (e.g., Song andHelmberger, 2007). As pointed out by Brocher (2005), it has been acommon practice in many regions that seismologists who do earth-quake hazard analyses first develop Vp models and then convert thoseto Vs models for two reasons: (1) There are much more Vp data thanVs data. (2) A Vs model is more important than a Vp model becauseshear and surface waves have much larger amplitudes, excite strongground motions, and consequently cause most of damage (Joyner,2000). During the last decade, S-wave velocity information becameincreasingly available through VSP analysis, full-waveform logging,multicomponent seismic data, and the analysis of teleseismic receiverfunctions. The non-unique interpretation of seismic data in terms ofrock type becomes inevitable when P- or S-wave velocities alone areavailable. Thus, the second objective of this investigation is toestablish some empirical relations between Vs and Vp and betweenlnVs and lnVp, and to characterize the variations of Poisson's ratio as afunction of seismic velocities Vp and Vs and elastic moduli for commonrock types. Such relationships are useful to constrain the full elasticproperties and characteristics of formation and deformation of rocksusing available Vp or Vs only. The scaling factor ∂lnVs/∂lnVp is ofparticular interest because it is used as a diagnostic for the origin ofheterogeneities in the Earth's interior since the factor, derived fromseismic inversions, is less ambiguous than the absolute values of theVp and Vs anomalies.

2. Data for statistic analysis

The data used for the statistic analysiswere taken from the samples onwhich both P- and S-wave velocities and density have been determined atrelatively high hydrostatic pressures (≥300 MPa) using the samelaboratory equipment and the samemethods (e.g., the pulse transmissiontechnique, Birch, 1960; Christensen, 1974; Kern, 1982). The confiningpressures are high enough to close the microcracks and pores with smallaspect ratios (width/length) so that the measured values can representthe intrinsic properties. The literature data were taken from Handbook ofSeismic Properties of Minerals, Rocks and Ores (Ji et al., 2002) andWanget al. (2005a,b) and Ji et al. (2007) and Ji et al. (2009). Only the velocitiesmeasured during depressurization were selected for the statistic studybecause the values correspond to the stable microstructural state (Burkeand Fountain, 1990; Ji et al., 1993; Wang et al., 2005a,b; Ji et al., 2007). Inorder to minimize the influence of seismic anisotropy, the arithmeticalmean P- and S-wave velocities were used for the computations of elasticmoduli for the bulk rock. The mean P-wave velocity of each sample iscalculated according to the following equation:

Vp = Vp Xð Þ + Vp Yð Þ + Vp Zð Þh i

= 3 ð1Þ

where Vp(X), Vp(Y) and Vp(Z) represent the P-wave velocities alongthe propagation directions X, Y and Z, respectively. The X direction is

parallel to the stretching lineation, the Y direction is perpendicular tolineation and parallel to foliation, and the Z direction is normal tofoliation. The mean S-wave of each sample is computed from theresults of six measurements that are designed XY, XZ, YX, YZ, ZX, andZY. The first letter refers to the propagation direction and the secondto the polarization direction.

Vs = Vs XYð Þ + Vs XZð Þ + Vs YXð Þ + Vs YZð Þ + Vs ZXð Þ + Vs ZYð Þ½ �= 6ð2Þ

The mean Vp and Vs values calculated from Eqs. (1) and (2) givevalues very close to true isotropic elastic properties even in highlyanisotropic rocks (Christensen and Ramananantoandro, 1971; Ji et al.,2003). The accuracy is estimated to be better than 0.5% for Vp and 1%for Vs (Christensen, 1985; Ji and Salisbury, 1993) and 2.5% forcalculated Poisson's ratio (Christensen, 1996). The measurements ofdensities of the samples, using the immersion method, have anaccuracy of ±0.005 g/cm3 at room conditions.

3. Linear relationship between Vp and Vs

Previous authors (e.g., Pickett, 1963; Castagna et al., 1985, 1993;Mavko et al., 1998) reported that the relationship between Vp and Vs

for sedimentary rocks (e.g., sandstone, limestone, dolomite, shale, andmudstone), which are generally porous, can be described by a linearcorrelation. Brocher (2005, 2008) plotted thousands of Vp and Vs datafor a wide variety of lithologies (from unconsolidated sediments tovery low porosity igneous rocks, and from non-welded volcanic tuffsto highly compact metamorphic rocks) all together on a single figureto draw a nonlinear fitting curve by eye. His empirical curve isdescribed by

Vp = 0:9409 + 2:0947Vs − 0:8206V2s + 0:2683V3

s − 0:0251V4s ð3Þ

or

Vs = 0:7858− 1:2344Vp + 0:7949V2p − 0:1238V3

p + 0:0064V4p ð4Þ

Eqs. (3) and (4) are valid for Vs between 0.3373 km/s and 5.0 km/sand Vp between 1.5 km/s and 8.5 km/s, respectively. Eq. (4) indicatesthat Vs=0.3373 km/s when Vp=1.5 km/s (In fresh water, P-wavetravels at about 1.5 km/s at 25 °C). The Vp and Vs data used forBrocher's regression fit, obtained from various methods (wireline

Fig. 1. Vs–Vp plots for peridotite (a), eclogite (b), mafic gneiss (c) and basalt (d) samples at 600 MPa. The experimental data are fitted using linear correlation.

Fig. 2. Vs–Vp plots for chalcopyrite (a), sphalerite (c) and pyrrhotite (d) samples at 600 MPa and pyrite (b) at 300 MPa. The experimental data are fitted using linear correlation.

63Q. Wang et al. / Tectonophysics 469 (2009) 61–72

Fig. 3. lnVs–lnVp plots for peridotite (a), eclogite (b), mafic gneiss (c) and basalt (d) samples at 600 MPa. The experimental data are fitted using linear correlation.

64 Q. Wang et al. / Tectonophysics 469 (2009) 61–72

borehole logs, vertical seismic profiles, laboratory or field measure-ments on hand samples at various effective confining pressures, in-situ estimates from seismic tomography studies, and regional velocitymodels) and are of different qualities, are extremely scattered.Furthermore, in Brocher's (2005, 2008) compilation, most of thevelocity measurements were made on samples from California, USA.

Here we show that the high-pressure Vp and Vs data of dryigneous and metamorphic rocks can be equally best fitted by thelinear correlation. A goodness-of-fit coefficient (R) is used as astatistical measure of how well the experimental data are fitted bythe empirical relation. R ranges between 0 and 1. If there is no fit atall, R is zero. A perfect fit gives a value of R=1.0. The higher R is,the better the fit is. The statistically fitting results for 12 commonlithologies (i.e., amphibolite, anorthosite, basalt, diorite, eclogite,felsic gneiss, gabbro-diabase, granite, intermediate gneiss, lime-stone, mafic gneiss, and peridotite) and 4 common sulfide ores (i.e.,chalcopyrite, pyrite, sphalerite and pyrrhotite) are summarized inTable 2.

Fig. 1a shows the Vs–Vp plots for 65 peridotites with variousdegrees of serpentinization, measured at a hydrostatic confiningpressure of 600 MPa. The least-squares linear fit gives Vs=0.728Vp

−1.380 (R=0.976) and Vp=1.310Vs+2.149 (R=0.976). Theserelations are valid when Vp ranges from 4.8 to 8.8 km/s and Vs

ranges from 2.2 to 5.0 km/s. The trends of velocity variation fromhigh to low values reflect essentially a progressive increase inserpentine content.

Fig. 1b shows the Vs–Vp variations for 55 eclogite samples at600 MPa. The linear fitting yield: Vs=0.496Vp+0.580 (R=0.789) inthe Vp range between 7.3 and 8.7 km/s, and Vp=1.257Vs+2.327(R=0.789) in the Vs range between 4.2 and 5.0 km/s. When only Vp

data exist, these Vs–Vp relations seem to give a reasonable estimatefor the average Vs values.

The linear regressions to Vs–Vp (Fig. 1c) and Vp–Vs data of 70mafic gneisses give: Vs=0.426Vp+0.924 (R=0.751) in the Vp

range from 6.5 to 7.6 km/s and Vp=1.325Vs+1.872 (R=0.751) inthe Vs range from 3.6 to 4.3 km/s, respectively. Although the linearcorrelation is clear, the data are somewhat scattered about thetrend line. This is not surprising in view of the highly variablecomposition and microstructure of the gneisses deformed andmetamorphosed under various conditions. Some points in Fig. 1cdeviated markedly from the regression lines, probably implyingthat these samples were inaccurately classified in term of lithology.

Fig. 1d plots 21 sets of seismic velocity data for basalts at ahydrostatic confining pressure of 600 MPa. The scattering of dataaround the trend line reflects mainly the variations in both porosityand pore geometry of the basalts. It is worthy to note that the smallpores with larger aspect-ratios (width/length) may remain atpressures as high as 600 MPa.

We also investigated the correlation between Vp and Vs for massiveoresof chalcopyrite, pyrite, pyrrhotite and sphalerite (Table2). Thesedatawere measured in Dr. Matt Salisbury's laboratory (Salisbury et al., 1996,1997, 2000) and incorporated in Handbook of Seismic Properties ofMinerals, Rocks and Ores (Ji et al., 2002). For chalcopyrite ores (Fig. 2a),Vs=1.032Vp−2.764 (R=0.913) in theVp range from5.0 to 6.6 km/s, andVp=0.807Vs+3.208 (R=0.913) in theVs range from2.5 to 4.2 km/s. Forpyrite ores (Fig. 2b), Vs=0.908Vp−2.111 (R=0.968) in the Vp rangefrom6.3 to 7.7 km/s, andVp=1.033Vs+2.623 (R=0.968) in theVs rangefrom 3.4 to 5.0 km/s. For sphalerite ores (Fig. 2c), Vs=0.704Vp−0.872(R=0.932) in the Vp range from 5.4 to 6.7 km/s, and Vp=1.235Vs+1.856 (R=0.932) in the Vs range from 2.7 to 4.2 km/s. For pyrrhotiteores (Fig. 2d), Vs=0.403Vp+0.965 (R=0.966) in the Vp range from 4.7to 6.5 km/s, andVp=2.319Vs−1.880 (R=0.966) in theVs range from2.7to 3.6 km/s. The above relations between Vp and Vs may help the miningindustry to search for deep massive sulfide ores using seismic methods.

Fig. 4. (a) Variation of Rs/p versus density for each lithology, (b) b–a plots for therelation Vs=aVp+b, and (c) b–a plots for the relation v=aVp+b.

65Q. Wang et al. / Tectonophysics 469 (2009) 61–72

4. Scaling factor Rs/p

Scaling factor Rs/p is defined as

Rs=p =AVs = Vs

AVp = Vp=

A lnVs

A lnVpð5Þ

where ∂Vs/Vs and ∂Vp/Vp are the S and P wave velocity anomalies,respectively. Clearly, Rs/p is a logarithmic ratio to describe the relativevariation in Vs and Vp for a given lithology. Rs/p can be estimated fromthe data of P wave travel time delay (∂tp) and S wave travel time delay(∂ts) at a given depth: Rs/p≈(Vs/Vp)(∂ts/∂tp). The velocity hetero-geneities in the Earth's interior can be caused by the variations intemperature, chemical composition or phase transformation. Rs/p hasbeen used as a diagnostic for the origin of heterogeneities in theEarth's interior because this factor, derived from seismic inversions, isless ambiguous than the absolute values of the anomalies (e.g.,Kennett et al., 1998; Karato and Karki, 2001). Robertson and Wood-house (1996) found a quasi-linear increase in Rs/p with depth from 1.7in the upper mantle to N2.5 in the lower mantle deeper than 2000 km.To our knowledge, the Rs/p values have not been systematicallyinvestigated for the common crustal rocks although these values arepotentially important constrains on the composition, physical stateand structure of the deep continental crust.

Fig. 3 illustrates the lnVs–lnVp plots for 65 peridotites, 55eclogites, 70 mafic gneisses, and 21 basalts at 600 MPa. The least-squares linear fit yields: lnVs=1.39lnVp−1.400 (R=0.981) forperidotite (Fig. 3a), lnVs=0.866lnVp−0.286 (R=0.782) for eclogite(Fig. 3b), lnVs=0.768lnVp−0.132 (R=0.756) for mafic gneiss (Fig.3c), and lnVs=1.096lnVp−0.755 (R=0.916) for basalt (Fig. 3d). Asshown in Fig. 4a, Rs/p varies systematically with lithology: 0.300 forgranite, 0.573 for diorite, 0.602 for felsic gneiss, 0.631 for inter-mediate gneiss, 0.721 for gabbro-diabase, 0.768 for mafic gneiss,0.866 for eclogite, 0.890 for amphibolite, 1.096 for basalt, 1.159 foranorthosite, and 1.391 for peridotite. Obviously, Rs/p increasesglobally with decreasing the content of SiO2 from acidic, intermedi-ate, through mafic to ultramafic rocks. Basalt displays a higher Rs/pvalue than gabbro-diabase although these mafic rocks have similarchemical compositions, indicating the effect of porosity (Table 2). Asporous basalt exists only in the uppermost part of the crust, Rs/pshould allow better constrains on lithological composition of thedeep continental crust that with the absolute value of Vp or Vs alone.Furthermore, a volumically important mass of eclogite that wasdelaminated from the lowermost part of the thickened continentalcrust and sunk into the upper mantle should be distinguished fromits surrounding peridotitic mantle in term of Rs/p. Otherwisediscrimination of eclogite from peridotite seems to be impossibleusing the absolute values of Vp or Vs.

Fig. 5 shows the lnVs–lnVp plots for massive ores. For chalcopyrite,pyrite, sphalerite and pyrrhotite lnVs=1.846lnVp−2.080 (R=0.924),lnVs=1.524lnVp−1.524 (R=0.968), lnVs=1.280lnVp−1.086(R=0.924), and lnVs=0.695lnVp−0.027 (R=0.964), respectively.Rs/p increases progressively from pyrrhotite (0.695), sphalerite (1.280),pyrite (1.524) to chalcopyrite (1.846), suggesting that the scaling factoris indeed a diagnostic for the chemical composition of sulfide ores.

Figs. 6 and 7 give two examples for the lnVc–lnVs (Vc is the bulkwave speed), lnρ–lnVs (ρ is the density) and lnρ–lnVp plots. Thelinear regression of these data yields the following logarithmicratios: Rc/s=∂lnVc/∂lnVs, Rρ/s=∂lnρ/∂lnVs, and Rρ/p=∂lnρ/∂lnVp. These ratios, which are usually derived from seismic andgeodynamical inversions, have been used jointly with Rs/p asdiagnostics for the origin of heterogeneities in the Earth's interior.Karato and Karki (2001) suggests that Rs/pN1.5 along with Rρ/sb0or Rρ/sN0 should indicate a dominant chemical or thermal origin ofthe mantle heterogeneity. For peridotite (Fig. 6), lnVc=0.515lnVs+1.039 (R=0.919), lnρ=0.443lnVs+0.528 (R=0.862), and

Fig. 5. lnVs–lnVp plots for chalcopyrite (a), sphalerite (c) and pyrrhotite (d) samples at 600 MPa and pyrite (b) at 300 MPa.

66 Q. Wang et al. / Tectonophysics 469 (2009) 61–72

lnρ=0.631lnVp−0.121 (R=0.865). Thus Rc/s=0.515, Rρ/s=0.443 and Rρ/p=0.631 for peridotite. For pyrrhotite (Fig. 7),lnVc=1.619lnVs−0.470 (R=0.922), lnρ=−1.359lnVs+2.977(R=0.935), and lnρ=−0.966lnVp+3.049 (R=0.922). HenceRc/s=1.619, Rρ/s=−1.359 and Rρ/p=−0.966 for pyrrhotite. It isof interest to note that different from pyrite and sphalerite,pyrrhotite and chalcopyrite have negative values for Rρ/s and Rρ/p.

5. Correlations of Poisson's ratio with Vs, Vp, G and E

For the rocks studied, there is generally a negative linearcorrelation between Poisson's ratio and Vs, G or E (Table 3 andFigs. 8 and 9). However, Poisson's ratio displays distinct trends ofvariation with Vp for different lithologies: υ decreases with increasingVp for anorthosite, basalt and peridotite, but increases with increasingVp for amphibolite, diorite, eclogite, felsic gneiss, gabbro-diabase,granite, intermediate gneiss, limestone, and mafic gneiss. Theincreases of υ with increasing Vp in general are not great butcontinuous.

Fig. 8 shows the variation in υ with Vs, Vp, G and E for 65peridotites at 600MPa. Poisson's ratio decreases linearly with increasein Vs, Vp, G and E. It is evident that υ displays a better correlation withVs (R=0.911) than Vp (R=0.802). Poisson's ratio also shows a lightlybetter correlation with G (R=0.901) than E (R=0.886). For granite,Poisson's ratio displays a decrease with increasing Vs but a slightincrease with increasing Vp. Poisson's ratio reveals also a slightdecrease with increase in G and E (Fig. 9). Similar correlationsbetween υ and Vs, Vp, G or E are observed for eclogite.

The linear-regression results for the sulfide ores (chalcopyrite,pyrite, pyrrhotite and sphalerite) are given in Table 3. Poisson's ratiodecreases linearly with increase in Vs, Vp, G and E for chalcopyrite,pyrite (Fig. 10), and sphalerite. For pyrrhotite (Fig. 11), however,

Poisson's ratio displays a trend of linear increase with increasing Vs,Vp, G and E. We hope that this new finding will encourage the searchfor the physical significance of these correlations.

The relation between Poisson's ratio (υ) and Vp is defined byυ=aVp+b, where a and b are two parameters to describe the slopeand intercept, respectively. It is surprising to see that in the b–adiagram (Fig. 4c) felsic rocks (i.e., granite, diorite, felsic gneiss andintermediate gneiss), mafic rocks (i.e., amphibolite, gabbro-diabase,eclogite, and mafic gneiss) and ultramafic rocks (i.e., peridotite) arecharacterized by high a and low b, moderate a and b, and low a andhigh b values, respectively. An opposite trend is observed for the Vs= f(Vp) linear function (Fig. 4b). The sulfide ores can be placed in thefollowing order of increasing the a-value and decreasing the b-value:chalcopyrite, pyrite, sphalerite and pyrrhotite when υ=aVp+b. Theexact reason for the above evolution remains unclear and is aninteresting theme for further study.

6. Discussion

It is generally described in the textbook of geophysics andmaterials science that Poisson's ratio depends on only the ratio ofVp/Vs and is insensitive to the absolute values of Vp and Vs (e.g.,Gercek, 2007). Such impression rises probably from the presenta-tion:

υ =12

1−Vp

Vs

� �2

−1

!−1" #ð6Þ

Here we have shown some obvious dependence of Poisson's ratioupon Vs and Vp for the common crustal and upper mantle rocks.

Fig. 6. lnVc–lnVs (a), lnρ–lnVs (b) and lnρ–lnVp plots for peridotite at 600 MPa. Vc is thebulk wave speed, and ρ is the density.

Fig. 7. lnVc–lnVs (a), lnρ–lnVs (b) and lnρ–lnVp plots for pyrrhotite at 600 MPa. Vc is thebulk wave speed, and ρ is the density.

67Q. Wang et al. / Tectonophysics 469 (2009) 61–72

Poisson's ratio decreases generally with increase in Vs, G and E.However, the dependence of υ on Vp seems to be complicated and tovary with lithology. At 600 MPa, amphibolite, diorite, eclogite, felsicgneiss, gabbro-diabase, granite, intermediate gneiss, limestone, andmafic gneiss (Table 3) display an increase in υ with increasing Vp.Anorthosite, basalt and peridotite, however, reveals a negativecorrelation between υ and Vp. Moreover, the sulfide ores can beclassified into two groups according to the υ–Vp relationship: anegative correlation for chalcopyrite, pyrite (Fig. 11) and sphalerite,whereas a positive correlation for pyrrhotite (Fig. 12).

Eq. (6) can be written as:

υ =V2p − 2V2

s

2 V2p − V2

s

� � ð7Þ

Eq. (7) indicates that υ is a function of both Vs and Vp. For a constantVp, υ always decreases nonlinearly with increase in Vs. For a constantVs, however, υ always increases nonlinearly with increase in Vp.

Differentiating Eq. (7) yields:

dυυ

= fdVp

Vp− dVs

Vs

!= f 1− Rs=p

� �dVp

Vpð8Þ

where f =2V2

pV2s

V2p − V2

s

� �V2p − 2V2

s

� � ð9Þ

As long as υN0, then (Vp2−2Vs

2)N0, and ζN0. Hence Eq. (8) showsthat the variation in Poisson's ratio depends on the relative variationrates in Vp and Vs. As long as the rate of increase in Vs is larger thanthat in Vp, dVs/VsNdVp/Vp, dv/vwill be negative and thus v decreaseswith increase of Vs and Vp (Kumar et al., 2003). Thus Eq. (8) indicatesa position correlation between Poisson's ratio and Vp (Table 3) whenRs/pb1 (e.g., amphibolite, diorite, eclogite, felsic gneiss, gabbro-diabase, granite, intermediate gneiss, limestone, mafic gneiss andpyrrhotite). A negative correlation occurs between Poisson's ratio

Table 3Parameters of the correlations between Poisson's ratio (υ) and Vs, Vp, G or E.

Lithology P (MPa) N υ=aVs+b υ=aVp+b υ=aG+b υ=aE+b

a b R a b R a b R a b R

RocksAmphibolite 600 29 −0.071 0.541 0.591 0.012 0.184 0.136 −0.002 0.354 0.420 −0.001 0.325 0.270Anorthosite 600 8 −0.058 0.516 0.719 −0.014 0.400 0.239 −0.002 0.373 0.562 −0.001 0.367 0.487Basalt 600 21 −0.045 0.431 0.569 −0.010 0.333 0.191 −0.002 0.331 0.528 −0.001 0.325 0.451Diorite 600 9 −0.042 0.093 0.240 0.054 −0.099 0.752 0.003 0.150 0.430 0.001 0.131 0.557Eclogite 600 55 −0.042 0.454 0.431 0.013 0.157 0.212 −0.001 0.338 0.422 −0.0003 0.319 0.300Felsic gneiss 600 67 −0.084 0.556 0.458 0.047 −0.050 0.443 −0.003 0.359 0.335 −0.0003 0.277 0.079Gabbro-diabase 600 47 −0.030 0.391 0.264 0.024 0.108 0.410 −0.001 0.314 0.261 −0.0002 0.298 0.150Granite 600 31 −0.174 0.880 0.761 0.085 −0.290 0.467 −0.007 0.497 0.642 −0.002 0.430 0.394Intermediate gneiss 600 14 −0.155 0.815 0.770 0.045 −0.043 0.257 −0.009 0.561 0.850 −0.004 0.596 0.716Limestone 600 26 −0.083 0.597 0.724 0.005 0.287 0.058 −0.004 0.443 0.732 −0.002 0.444 0.661Mafic gneiss 600 70 −0.043 0.446 0.383 0.021 0.131 0.321 −0.001 0.323 0.294 −0.0002 0.304 0.173Peridotite 600 65 −0.046 0.476 0.911 −0.030 0.516 0.802 −0.002 0.380 0.901 −0.001 0.382 0.886

Sulfide oresChalcopyrite 600 11 −0.117 0.649 0.942 −0.101 0.860 0.724 −0.007 0.532 0.894 −0.003 0.580 0.826Pyrite 300 12 −0.080 0.549 0.914 −0.064 0.660 0.784 −0.001 0.320 0.856 −0.001 0.325 0.829Pyrrhotite 600 24 0.084 −0.023 0.668 0.044 0.004 0.834 0.005 0.043 0.414 0.003 −0.018 0.659Sphalerite 600 9 −0.060 0.472 0.730 −0.027 0.433 0.434 −0.001 0.313 0.378 −0.0003 0.308 0.315

68 Q. Wang et al. / Tectonophysics 469 (2009) 61–72

and Vp (Table 3) when Rs/pN1 (e.g., anorthosite, basalt, peridotite,chalcopyrite, pyrite and sphalerite).

υ =E2G

− 1 ð10Þ

we obtain:

dυυ

=1

1− 2GE

� � dEE

− dGG

� �ð11Þ

Because 2G/E=1/(v+1) and 0≤v≤0.5, 2G/E≤1. From Eq. (11), wecan easily observe that if the rate of G increase is larger than that of Eincrease, dv/v will be negative and hence v will vary in the oppositedirection as those of E and G. For the rocks investigated, v generallydisplays a negative correlation with G and E, indicating that dG/G iscommonly larger than dE/E.

However, Vp and Vs are always mutually correlated and both tend tovary in the samedirection (increase or decrease). Thus, the variation of vwith Vs and Vp may be complicated: v may monotonically increase,monotonically decrease, first increase and then decrease (convex line),or first decrease and then increase (concave line) with increase in Vs orVp.

Considering that Vp is linearly correlated with Vs:

Vp = aVs + b ð12ÞdυdVs

= −b a + b

Vs

� �a + b

Vs

� �2−1� 2

V2s

ð13Þ

Poisson's ratio decreases with increasing Vs if dv/dVs calculatedfrom Eq. (13) is negative. For example, the calculated dv/dVs values forperidotite (Fig. 12a) and granite (Fig.12c) are always negative in the Vs

ranges of experimental measurements (2.2–5.0 km/s for peridotiteand 3.4–3.9 km/s for granite). These negative values of dv/dVs

correspond to the continuous decrease in v with increasing Vs forperidotite (Fig. 8a) and granite (Fig. 9a).

When Vs is correlated with Vp,

Vs = aVp + b ð14Þ

dυdVp

=

Vp

b + aVpð Þ2 1− aVp

b + aVpð Þ� �Vp

b + aVp

� �2−1� �2 ð15Þ

Poisson's ratio decreases with increasing Vp if dv/dVp calculated fromEq. (15) is negative. In the case of peridotite (Fig. 12b), for example,the calculated dv/dVp values are negative in the Vp range ofexperimental measurements (4.8–8.8 km/s). This explains whyPoisson's ratio of peridotite displays a clear trend of decrease withincreasing Vp (Fig. 8b). For granite (Fig. 12d), the calculated dv/dVp

values are positive in the Vp range from 6.0–6.5 km/s, correspondingto an increase in v with increasing Vp (Fig. 9b).

7. Conclusions

Data from high-pressure laboratory measurements of Vp and Vs

are statistically analyzed for 12 common categories of rocks (i.e.,amphibolite, anorthosite, basalt, diorite, eclogite, felsic gneiss, gabbro-diabase, granite, intermediate gneiss, limestone, mafic gneiss, andperidotite) and 4 types of massive sulfide ores (i.e., chalcopyrite,pyrite, pyrrhotite, and sphalerite). The analysis shows that the linearcorrelation provides rather good descriptions for the relationshipsbetween Vs and Vp or lnVs and lnVp. These empirical relations can givea reasonable estimate for the Vs value when only Vp data exist or viseverso. The logarithmic ratio Rs/p (i.e., ∂lnVs/∂lnVp) varies system-atically with lithology: 0.300 for granite, 0.573 for diorite, 0.602 forfelsic gneiss, 0.631 for intermediate gneiss, 0.721 for gabbro-diabase,0.768 for mafic gneiss, 0.866 for eclogite, 0.890 for amphibolite, and1.391 for peridotite. The trend of an obvious increase in Rs/p withdecreasing the content of SiO2 from acidic to intermediate, thenthrough mafic to ultramafic rocks suggests that the ratio can be usedas a diagnostic for the composition of the deep continental crust withless ambiguity than the absolute values of Vp and Vs. It is of particularinterest that eclogite (0.866) displays a significantly lower Rs/p valuethan peridotite (1.391). An eclogitic mass that was delaminated fromthe lowermost part of the thickened continental crust and sunk intothe upper mantle, if its volume is sufficiently large, should bedistinguishable from its surrounding peridotitic mantle in term ofRs/p. Otherwise the discrimination of eclogite from peridotite seems tobe impossible using the absolute values of Vp or Vs alone.

Poisson's ratio,which is believed to providemuch tighter constraintson lithological composition than either Vp or Vs alone, is found to belinearly correlatedwith Vs, Vp,G and E for the common rocks and sulfideores. Adecrease inPoisson's ratio is commonlyassociatedwith increasesin Vs, G and E. The variation of Poisson's ratio with Vp depends on thescaling factor Rs/p. The positive and negative corrections betweenPoisson's ratio and Vp occur when Rs/pb1 and Rs/pN1, respectively. It isalso found that different from pyrite and sphalerite, pyrrhotite and

Fig. 9. Variations in Poisson's ratio with Vs (a), Vp (b), G (c) and E (d) for granite at 600 MPa.

Fig. 8. Variations in Poisson's ratio with Vs (a), Vp (b), G (c) and E (d) for peridotite at 600 MPa.

69Q. Wang et al. / Tectonophysics 469 (2009) 61–72

Fig. 11. Variations in Poisson's ratio with Vs (a), Vp (b), G (c) and E (d) for pyrrhotite ore at 600 MPa.

Fig. 10. Variations in Poisson's ratio with Vs (a), Vp (b), G (c) and E (d) for pyrite ore at 300 MPa.

70 Q. Wang et al. / Tectonophysics 469 (2009) 61–72

Fig. 12. Variations in dv/dVs with Vs (a and c) and dv/dVp with Vp (b and d) for peridotite (a–b) and granite (c–d).

71Q. Wang et al. / Tectonophysics 469 (2009) 61–72

chalcopyrite have negative values for Rρ/s (∂lnρ/∂lnVs) and Rρ/p (∂lnρ/∂lnVp). Interesting theoretical problems arise to explain the resultspresented in the paper.

Acknowledgements

We thank the Natural Sciences and Engineering Council of Canadaand the Chinese Academy of Sciences for research grants. We aregrateful to Dr. Matt Salisbury for his seismic velocity data of sulfides,and Drs. H. Thybo, H. Kern and an anonymous reviewer forconstructive comments.

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