the relationships between large scale variations in shear velocity density and compressional velocity in the earth s mantle a large data set of surface wave phase anomalies body wave travel times normal mode splitting functions and long period waveforms is used to investigate the scaling between shear velocity density and compressional velocity in the earth s mantle we introduce a methodology that allows construction of joint models with various levels of scaling complexity d ln d lnvs d lnvs d lnvp in order to detect seismological signatures of chemical heterogeneity we demonstrate that the data sets considered cannot be fit concurrently with a uniform or a positive and uniform throughout the mantle the variance reductions to p wave travel times and vp sensitive modes are up to 40 higher with our preferred model of anisotropic shear and compressional velocity than the recent anisotropic shear velocity model s362ani m which was constructed assuming a uniform throughout the mantle several features reported in earlier tomographic studies persist after the inclusion of new and larger data sets anticorrelation between bulk sound and shear velocities in the lowermost mantle as well as an increase in with depth in the lower mantle are largely independent of the regularization scheme when correlations between density and shear velocity variations are imposed in the lowermost mantle variance reductions of several spheroidal and toroidal modes deteriorate by as much as 40 recent measurements of the splitting of 0s2 in particular are largely incompatible with perfectly correlated shear velocity and density heterogeneity throughout the mantle a way to significantly improve the fits to various data sets is by allowing independent density perturbations in the lowermost mantle our preferred joint model consists of denser than average anomalies 1 peak to peak at the base of the mantle roughly coincident with the low velocity superplumes the relative variation of shear velocity density and compressional velocity in our study disfavors a purely thermal contribution to heterogeneity in the lowermost mantle with implications for the long term stability and evolution of superplumes

mantle tomography and its relation to temperature and composition we propose a new method to constrain lateral variations of temperature and composition in the lower mantle from global tomographic models of shear and compressional wave speed we assume that the mantle consists of a mixture of perovskite and magnesio wustite in a first stage we directly invert vp and vs anomalies for variations of temperature and composition using the appropriate partial derivatives or sensitivities of velocities to temperature and composition however uncertainties in the tomographic models and in the sensitivities are such that variations in composition are completely unconstrained inferring deterministic distributions of temperature and composition being currently not possible we turn to a statistical approach which allows to infer several robust features comparison between synthetic and predicted ratios of the relative shear to compressional velocity anomalies indicates that the origin of seismic anomalies cannot be purely thermal but do not constrain the amplitude of the variations of temperature and composition we show that we can estimate these variations using histograms of the relative vp and vs anomalies at a given depth we computed histograms for a large variety of cases and found that at the bottom of the mantle variations in the volumic fraction of perovskite from 14 to 10 are essential to explain seismic tomography in the mid mantle anomalies of perovskite are not required but moderate variations up to 6 can explain the observed distributions equally well these trade offs between anomalies of temperature and composition cannot be resolved by relative velocity anomalies alone an accurate determination of temperature and composition requires the knowledge of density variations as well we show that anomalies of iron can then also be resolved 2003 elsevier b v all rights reserved

low angle range flank faults in the panamint inyo and slate ranges california implications for recent tectonics of the death valley region four of the mountain ranges near death valley california display exhumed low angle normal faults on their flanks features originally referred to as turtleback structures these fault surfaces are smooth planar to curviplanar and defined by faceted spurs and coincident interfluve crests the lowest parts of these faults are overlain in tectonic contact by poorly indurated fanglomerate and locally by quaternary volcanic rocks steep fault scarps in alluvium are present at the foot of each turtleback type flank although these may be examples of active low angle normal faults it is difficult to establish the dip histories of these faults the problem with constraining the dip histories is one of missing and concealed information because the exposed hanging wall rocks are some of the youngest material cut and displaced by the currently shallow dipping faults another important relationship along these range flanks is the intersection between the low angle faults and steep scarp forming faults that cut alluvium at the range fronts although this relationship remains unconstrained in most of the ranges a shallow dipping fault is cut by one of the steep faults in panamint valley suggesting that in one area at least the low angle faults are not active the low angle faults and not the steep neotectonic faults are probably responsible for most of the opening of the present valleys because each low angle fault intersects the valley floor at the range front none of the low angle faults are perched high in the ranges as would be the case if the steep faults had uplifted them significantly therefore even if there has been a geologically recent transition from low angle normal faulting to steep normal with or without strike slip faulting the modern topography still reflects the former regime strictly speaking the death valley region may not be an example of active low angle normal faults and supradetachment basins

a new catalogue of normal mode splitting function measurements up to 10 mhz the splitting of the earth s free oscillation spectra places important constraints on the wave speed and density structure of the earth s mantle and core we present a new set of 164 self coupled and 32 cross coupled splitting functions they are derived from modal spectra up to 10 mhz for 91 events with mw 7 4 from the last 34 yr 1976 2010 our data include the 2001 june 23 peru event mw 8 4 the sumatra events of 2004 mw 9 0 and 2005 mw 8 6 the 2008 wenchuan china event mw 7 9 and the 2010 chile event mw 8 8 the new events provide significant improvement of data coverage particularly in continental areas almost half of the splitting functions have never been measured before in particular we measured 33 new modes sensitive to mantle compressional wave velocity 10 new inner core sensitive modes and 22 new cross coupled splitting functions these provide new constraints on the large scale compressional structure of the mantle and the odd degree structure of the mantle and inner core and can be used in future inversions of heterogeneous earth structure our new splitting function coefficient data set will be available online

reliable mantle density error bars an application of the neighbourhood algorithm to normal mode and surface wave data summary we demonstrate the feasibility and reliability of a new approach to seismic modelling of long wavelength mantle structure in doing so we present new estimates of mantle model uncertainties that successfully explain apparent discrepancies in earlier 3d seismic density models and justify a new generation of model building the characteristics of good models and of modelling error are described by model space maps that display the data fit of a representative set of potential models the neighbourhood algorithm na recently developed by malcolm sambridge allows efficient production of such maps for the long wavelength model parametrizations used in imaging the mantle density using normal mode data we observe that when na is applied to our global modelling problem the results are not only self consistent and consistent with independent resolution tests but also explain discrepancies in the results of damped inversions synthetic mapping experiments using the sensitivity kernels and measurement errors of several data catalogues allow us to determine the resolution of these data sets such resolution tests let us know which model parametrizations and data sets if any can yield meaningful results in inversions of real data the new resolution tests reveal that recent density models from damped inversions of normal mode data are not robust however we also show that the addition of the most recent long wavelength data will for the first time reduce model covariances enough to achieve robust constraints on the depth and size of long wavelength density heterogeneity throughout the mantle

calculation of normal mode spectra in laterally heterogeneous earth models using an iterative direct solution method summary normal mode observations play an important role in studying broad scale lateral variations in the earth such studies require the calculation of accurate synthetic spectra in realistic earth models and this remains a computationally challenging problem here we describe a new implementation of the direct solution method for calculating normal mode spectra in laterally heterogeneous earth models in this iterative direct solution method the modecoupling equations are solved in the frequency domain using the preconditioned biconjugate gradientalgorithm andthetime domainsolutionisrecoveredusinganumericalinversefourier transform a number of example calculations are presented to demonstrate the accuracy and efficiency of the method for performing large full coupling calculations as compared to methods based on matrix diagonalization and the traditional direct solution method

gypsum a joint tomographic model of mantle density and seismic wave speeds 1 gypsum is a 3 d model of mantle shear wave s speeds compressional wave p speeds and density the model is developed through simultaneous inversion of seismic body wave travel times p and s and geodynamic observations while using realistic mineral physics parameters linking wave speeds and density geodynamic observations include the global free air gravity field divergence of the tectonic plates dynamic topography of the free surface and the flow induced excess ellipticity of the core mantle boundary gypsum is built with the philosophy that heterogeneity that most closely resembles thermal variations is the simplest possible solution models of the density field from earth s free oscillations have provided great insight into the density configuration of the mantle but are limited to very long wavelength solutions alternatively scaling higher resolution seismic images to obtain density anomalies generates density fields that do not satisfy geodynamic observations the current study provides a 3 d density model for the mantle that directly satisfies geodynamic and seismic observations through a joint seismic geodynamic inversion process notable density field observations include high density piles at the base of superplume structures supporting the general results of past normal mode studies however we find that these features are more localized and have lower amplitude than past studies would suggest when we consider both fast and slow seismic anomalies in gypsum we find that p and s wave speeds are strongly correlated throughout the mantle however we find a low correlation of fast s wave zones in the deep mantle 1500 km depth with the corresponding p wave anomalies suggesting a systematic divergence from simplified thermal effects in ancient subducted slab anomalies the cratonic lithosphere and d regions are shown to have strong compositional signatures however we argue that temperature variations are the primary cause of p wave speed s wave speed and density anomalies throughout most of the mantle

bayesian inference of mantle viscosity from whole mantle density models the long wavelength geoid is sensitive to earth s mantle density structure as well as radial variations in mantle viscosity we present a suite of inversions for the radial viscosity profile using

deep mantle seismic modeling and imaging detailed seismic modeling and imaging of earth s deep interior is providing key information about lower mantle structures and processes including heat flow across the core mantle boundary the configuration of mantle upwellings and downwellings phase equilibria and transport properties of deep mantle materials and mechanisms of core mantle coupling multichannel seismic wave analysis methods that provide the highest resolution deep mantle structural information include network waveform modeling and stacking array processing and 3d migrations of p and s wave seismograms these methods detect and identify weak signals from structures that cannot be resolved by global seismic tomography some methods are adapted from oil exploration seismology but all are constrained by the source and receiver distributions long travel paths and strong attenuation experienced by seismic waves that penetrate to the deep mantle large and small scale structures with velocity variations ranging from a fraction of a percent to tens of percent have been detected and are guiding geophysicists to new perspectives of thermochemical mantle convection and evolution

normal mode and free air gravity constraints on lateral variations in velocity and density of earth s mantle with the use of a large collection of free oscillation data and additional constraints imposed by the free air gravity anomaly lateral variations in shear velocity compressional velocity and density within the mantle dynamic topography on the free surface and topography on the 660 km discontinuity and the core mantle boundary were determined the velocity models are consistent with existing models based on travel time and waveform inversions in the lowermost mantle near the core mantle boundary denser than average material is found beneath regions of upwellings centered on the pacific ocean and africa that are characterized by slow shear velocities these anomalies suggest the existence of compositional heterogeneity near the core mantle boundary

mapping the upper mantle three dimensional modeling of earth structure by inversion of seismic waveforms a method is presented for the inversion of waveform data for the three dimensional distribution of seismic wave velocities the method is applied to data from the global digital networks international deployment of accelerometers global digital seismograph network the selected data set consists of some 2000 seismograms corresponding to 53 events and 870 paths the moment tensors of the earthquakes are determined through an iterative procedure which minimizes the corrupting influence of lateral heterogeneity a global model is constructed for shear wave velocity expanded up to degree and order 8 in spherical harmonics and described by a cubic polynomial in depth for the upper 670 km of the earth s mantle although no a priori information is incorporated the model predictions reproduce much of what is known about the dispersion of mantle waves for example high phase velocities for shields low velocities at ridges and a strong degree 2 pattern for rayleigh waves since the method makes use of complete waveforms overtone data are also included it is shown that the model is reproducible in that substantially the same model can be constructed from each half of the total data set considered independently the model shows that shields and ridges are major features in the depth interval 25 250 km the ridges of the southern pacific and the larger shields persist to 350 km but the southeast indian rise is underlain by a high velocity anomaly at this depth as is much of the mid atlantic ridge at 450 650 km the major features are a broad region of high velocities incorporating south america much of the south atlantic and parts of west africa a broad region of low velocities in the central and eastern pacific high velocities in the western pacific and a low velocity anomaly beneath the red sea and the gulf of aden in the absence of a crustal correction degrees 2 and 3 show a high positive correlation with the geoid paradoxically this is largely destroyed when the distribution in crustal thickness is taken into account spherical harmonic degrees 4 7 show a significant negative correlation

compressible thermochemical convection and application to lower mantle structures a new finite element code for compressible thermochemical convection is developed to study the stability of a chemical layer at the base of the mantle using composition dependent compressibility and a density difference between compositions at a reference pressure a composition dependent density profile is derived together with depth dependent thermal expansion this combination of parameters yields a wide range of dynamic evolutions for the chemical layer the chemical structures are classified into five major categories classical plumes mushroom shaped plume domes ridges and continuous layers and a few abnormal cases such as hourglass shaped plumes and columnar plumes several models have a chemical structure morphologically similar to the african low v s structure in the lower mantle at least at a single time guided by our models several dynamic scenarios are proposed for the dynamic nature of the lower mantle low velocity structures a k a superplumes including plumes at an early stage plume clusters ridges passive piles sluggish domes and high bulk modulus domes we predict seismic velocity anomalies from these dynamic models the thermoelastic parameters used in the conversion are additional constraints we compare the density structure with normal mode inversion the predicted seismic signature observations and the required thermoelastic parameters with mineral physics data among the proposed scenarios only the scenario of high bulk modulus domes satisfies all constraints simultaneously the implication on the geochemistry and mineralogy of lower mantle chemical structures is discussed

synthetic inversions for density using seismic and gravity data density variations drive mass transport in the earth from plate tectonics to convection in the mantle and core nevertheless density remains poorly known because most geophysical measurements used to probe the earth s interior either have little sensitivity to density suffer from trade offs or from non uniqueness with the ongoing expansion of computational power it has become possible to accurately model complete seismic wavefields in a 3 d heterogeneous earth and to develop waveform inversion techniques that account for complicated wavefield effects this may help to improve resolution of density here we present a pilot study where we explore the extent to which waveform inversion may be used to better recover density as a separate independent parameter we perform numerical simulations in 2 d to investigate under which conditions and to what extent density anomalies may be recovered in the earth s mantle we conclude that density can indeed be constrained by seismic waveforms mainly as a result of scattering effects at density contrasts as a consequence the low frequency part of the wavefield is the most important for constraining the actual extent of anomalies while the impact of density heterogeneities on the wavefield is small compared to the effects of velocity variations it is likely to be detectable in modern regional to global scale measurements we also conclude that the use of gravity data as additional information does not help to further improve the recovery of density anomalies unless strong a priori constraints on the geometry of density variations are applied this is a result of the inherent physical non uniqueness of potential field inverse problems finally in the limited numerical setup that we employ we find that the initially supplied anomalies in s and p velocity models are of minor importance

on the resolution of density anomalies in the earth s mantle using spectral fitting of normal mode data summary the resolution of the 3 d density structure in the earth s deep interior has long eluded geoscientists high quality data from digital seismic instruments emplaced this decade have renewed interest in the measurement of low frequency earth normal modes with the goal of extracting heterogeneous density structure here we perform a series of synthetic experiments aimed at investigating the resolution of lateral variations in the mantle from normal mode spectral data contamination effects between seismic velocities and density are examined in two ways 1 by using resolution matrices computed from data kernels and 2 by inverting synthetic spectra computed from realistic input earth models the first type of experiment assumes that the non linearity of the inverse problem is weak no such assumptions are necessary in the second type of tests which nevertheless concur in their results with the former these synthetic tests indicate that density structure retrieved from presently available normal mode data is not reliable contamination of seismic velocity structure into the density model space produces patterns that resemble those obtained from inversions of real data the resulting density models appear to be dependent on a combination of the starting velocity models and the model parametrization

whole earth structure estimated from seismic arrival times the arrival times of 10 compressional and shear phases p s pp pcp ss scs pk pab pk pbc pk pdf sk sac and two sets of differential times ss s410s and ss s660s are used to infer the earth s three dimensional structure with the exception of ss sds the raw data are composed of arrival times from the international seismological centre isc catalogue the isc events have been relocated using p and s phases and the ak135 radial earth model a total of 818 655 summary arrival times constrain lateral velocity variations in the crust mantle and core we also determine relocation parameters for all events stations corrections for each phase boundary topography 410 km 660 km core mantle boundary inner core boundary and inner core anisotropy variations our model of p and s velocity variations in the crust and mantle contains the general features observed in other body wave inversions a low velocity ring around the pacific ocean basin in the depth range 35 400 km high velocity anomalies associated with subducting slabs most prominent in the depth range 660 1070 km and high velocity anomalies surrounding the pacific basin just above the core mantle boundary in the outer core there is evidence of a latitudinal variation in velocity approaching 0 6 in magnitude with lower p velocities at the poles and higher velocities near the equator this pattern is observed throughout the outer core but is most prominent in the depth range 3671 4071 km and near the inner core boundary synthetic tests indicate that outer core features are moderately resolvable and inversions of random arrival times indicates that the anomalies are a factor of 2 above the estimated parameter noise level lateral heterogeneity in the inner core trades off with anisotropy resulting in poor resolution of isotropic velocity variations anisotropy in the inner core is predominantly subparallel to the earth s axis and of the order of 3 but this value can decrease if more inner core isotropic lateral heterogeneity is allowed a comparison of structure above and below the core mantle boundary cmb indicates that velocity heterogeneity is uncorrelated across the cmb but there is a moderate correlation between cmb topograpy and outer core heterogeneity an inversion in which the pk pdf residuals were absent indicates that the latitudinal pattern of outer core velocity variations is not due to the influence of inner core velocity structure

seismic heterogeneity in the mantle strong shear wave signature of slabs from joint tomography abstract the primary source of information on heterogeneity within the earth comes from seismic tomography a powerful tool for examining the character of heterogeneity comes from the comparison of images of bulk sound and shear wavespeed extracted in a single inversion since this isolates the dependencies on the elastic moduli however particularly in such multi parameter inversions there are many hidden facets which can have a strong influence on the results such as the weightings between parameters and in the misfit functions joint inversion with restricted data sets giving comparable cover for p and s waves provides useful checks on more inclusive studies and can provide relatively high resolution in some areas the relative behaviour of bulk sound and shear wavespeed can provide a useful guide to the definition of heterogeneity regimes for subduction zones a large part of the tomographic signal comes from s wavespeed variations in the upper mantle and transition there can be significant bulk sound speed contributions for younger slabs and in stagnant slabs associated with slab roll back for subducted oceanic lithosphere older than about 90 ma shear wavespeed variations nearly always are dominant and so the p wave images are controlled by shear modulus variations the narrow segments of fast wavespeeds in the depth range 900 1500 km in the lower mantle are dominated by s variations with very little bulk sound contribution deep in the mantle there are many fast features without obvious association with subduction in the last 100 ma which suggests long lived preservation of components of the geodynamic cycle

using probabilistic seismic tomography to test mantle velocity density relationships we use a neighborhood algorithm to explore the fit to long period seismic data of a wide variety of long wavelength mantle models this approach to the global tomographic inverse problem yields probability distributions for seismic velocities density and related properties as functions of depth such distributions can be robust even when individual models are not and allow us to test several assumptions about the earth that have long been enforced a priori in inversions in particular we are able to test the paradigm ofdeep mantle heterogeneity that is dominantly thermal in origin producing velocity and density perturbations that are well correlated and have relative amplitudes given by nlnb nlnvs 6 0 5 our distributions show that such relationships are unlikely and even though the results are consistent with recent best fitting models from damped seismic inversions they demonstrate that many specific properties ofsuch models are not robust the data clearly f density perturbations that are poorly or negatively correlated with velocity heterogeneity and have amplitudes several times larger yielding nlnb nlnvs s 1 0 than damped inversions allow these characteristics are most pronounced in the upper mantle transition zone and the base ofthe lower mantle suggesting layered convection the negative density velocity correlations f at these depths imply dominantly chemical heterogeneity while the likelihood ofrelatively high amplitude density variations suggests that variable iron content is an important component ofthis heterogeneity these results which we show to be consistent with independent gravity constraints represent a profound change in the interpretation of seismic constraints in addition the distributions show that even though best fitting density models from recent inversions or our sampling are consistent with the data most specific properties of such models are not robust this implies that it is more appropriate to use seismic model distributions rather than individual models to make geodynamic and geochemical inferences

can we resolve 3d density heterogeneity in the lower mantle we reexamine the possibility of resolving 3d density structure in the mantle by performing test inversions using a dataset of well constrained degree 2 splitting coefficients we show that degree 2 in vs and vp can be resolved and confirm that dlnvs dlnvp 2 5 at depths 2000km indicating a chemical component to large scale heterogeneity models with positively correlated with vs in the deepest mantle fit the data at least as well as those in which density and velocities are anti correlated however some rough bounds on dln dlnvs can be inferred models obtained from geodynamic considerations are acceptable in contrast to models with large negative 0 6 dln dlnvs throughout the lower mantle the most robust feature is a marked increase in dln dlnvs at the top of the lower mantle reaching a maximum around 1500 km

partial melting in a thermo chemical boundary layer at the base of the mantle abstract seismological detections of complex structures in the lowermost mantle boundary layer the d region motivate a conceptual model of a compositionally stratified thermo chemical boundary layer tcbl within which lateral temperature variations sustained by large scale mid mantle flow cause variations of partial melt fraction partial melt fractions of from 0 to 30 in the tcbl occur due to the eutectic of the boundary layer material lying below the high temperature at the core mantle boundary cmb coupled with the presence of steep thermal gradients across the boundary layer regions of the tcbl with the highest temperatures have extensive partial melting producing lateral chemical variations and strong effects on seismic velocities and boundary layer dynamics this tcbl concept provides a relatively simple framework that can plausibly account for diverse seismological observations such as predominance of large scale volumetric elastic wave velocity heterogeneity in the boundary layer laterally extensive but intermittent abrupt shear and compressional velocity increases and decreases at the top of the d region small scale topography of d velocity discontinuities thin ultra low velocity zones at the cmb widespread shear wave anisotropy in d bulk sound velocity anomalies detected in low shear velocity regions of d and large scale upwellings from the most extensively melted regions of the boundary layer neutral or negative buoyancy of the partial melt in the tcbl is required along with an increase in bulk modulus and some density increase of the boundary layer material relative to the overlying mantle more complex models in which the partially melted chemical boundary layer is laterally displaced by mid mantle downwellings are also viable but these appear to require additional special circumstances such as a phase change efficient segregation of slab crustal material or abrupt onset of anisotropy in order to account for rapid velocity increases at the top of d the precise nature of the compositional anomaly or anomalies in d and the eutectic composition of this zone are yet to be determined but it appears likely that partial melting within the lowermost mantle plays a paramount role in the observed seismic velocity heterogeneity increased resolution geodynamic calculations of dense boundary layer structures with partial melting and attendant viscosity reductions and chemical heterogeneity are needed to quantify this tcbl concept

global mantle shear velocity model developed using nonlinear asymptotic coupling theory we present a three dimensional shear velocity model of the whole mantle developed using s h waveform data the model is expressed horizontally in terms of spherical harmonics up to degree 12 and vertically in terms of legendre polynomials up to degrees 5 and 7 in the upper and lower mantle respectively what distinguishes this model from other tomographic models published to date is 1 the theoretical normal mode based wave propagation approach where we include across branch mode coupling terms in order to model the body wave sensitivity to structure along the path more accurately 2 the wave packet weighting scheme which allows to balance contributions from high amplitude and low amplitude phases increasing the resolution in some parts of the mantle we also relax the constraints on the moho depth which is allowed to vary in the inversion thus absorbing some uncertainties in crustal structure the resulting model is generally in good agreement with other recent global mantle s velocity models and with some regional models the rms profile with depth has more power than other models in the upper mantle lower mantle transition region and the zone of increased power and low degree structure near the base of the mantle is confined to the last 500 km in depth this model provides a particularly good fit to the non hydrostatic geoid through harmonic degree 12 79 variance reduction as well as good fits to observed splitting functions of s velocity sensitive mantle modes indicating that both large scale and small scale features are really well constrained

even degree lateral variations in the earth s mantle constrained by free oscillations and the free air gravity anomaly the recent occurrence of several large earthquakes in particular the 1994 june 9 bolivia event has motivated a re examination of the earth s large scale heterogeneity from a normal mode or free oscillation perspective compared to earlier studies the number of normal mode constraints on lateral variations in the mantle has increased five fold and toroidal and cross coupled modes complement the traditional spheroidal mode data set it is demonstrated that this large collection of mode data combined with the free air gravity anomaly can reliably constrain even degree lateral variations in wave velocities as well as density we present the first whole mantle density model constrained by seismology our shear and compressional velocity models are consistent with existing models based upon traveltimes and waveforms and are reasonably well correlated throughout the mantle shear and bulk sound velocity models exhibit a gradual decrease in correlation with depth and are anti correlated near the core mantle boundary we find that lateral variations in density are poorly correlated with wave velocities and are locally anti correlated with shear velocity in the lowermost mantle the correlations between wave velocities and density suggest both a thermal and a compositional origin to lateral heterogeneity in addition to traditional maps of lateral variations in wave velocity we also present maps of lateral variations in shear and bulk moduli the inversion puts weak constraints on even degree topographic variations on the core mantle boundary the 660 km discontinuity and dynamic free surface topography finally we determine both radially and laterally varying scaling relationships including poisson s ratio