Experimental Studies on Elasticities of Forsterite-Enstatite Composites:Towards an Understanding of the Rules of Mixture

Which rule of mixture is the best for predicting the overall elastic properties of polyphase rocks based on the elastic properties and volume fractions of their constituents? In order to address this question, we sintered forsterite-enstatite polycrystalline aggregates with a varied forsterite volume fraction (0, 0.2, 0.4, 0.5, 0.6, 0.8, and 1.0). Elastic properties (shear, bulk, and Young's moduli) of these synthesized composites were measured as a function of pressure up to 3.0 GPa in a liquid-medium piston-cylinder apparatus using a high-precision ultrasonic interferometric technique. The experimental data can be much better described by the shear-lag model than by the commonly used simple models such as Voigt, Ruess and Hill averages, Hashin-Shtrikman bounds, Ravichandran bounds, Halpin-Tsai equations, and Paul's calculations. We attributed this to the fact that the elastic interaction and stress transfer between phases are neglected in all the models except for the shear-lag model. In particular, t     

多重孔岩石微分等效介质模型及其干燥情形下的解析近似式

经典的微分等效介质(DEM)理论可用于确定多孔介质的弹性性质,但由于缺乏多重孔DEM方程,其估计的多重孔岩石的等效弹性模量依赖于包裹体(即不同孔隙纵横比的孔或缝)的添加顺序.本文首先从Kuster-Toksöz理论出发建立了Zimmermann和Norris两种形式的多重孔DEM方程.Norris形式的多重孔DEM方程预测的等效弹性模量总是位于Hashin-Shtrikman上下限内,而Zimmermann形式的多重孔DEM方程有时会越界.然后,通过使用干燥岩石模量比的解析近似式,对两个相互耦合的Norris形式DEM方程进行解耦得到干燥多重孔岩石的体积和剪切模量解析式.用全DEM方程的数值解对解析近似式的有效性进行了测试,解析公式的计算结果在整个孔隙度分布区间与数值解吻合良好.对实验室测量数据在假设岩石含有双重孔隙的情形下用双重孔DEM解析公式对岩石的弹性模量进行了预测,结果表明,解析式准确地预测了弹性模量随孔隙度的变化.双重孔(即软、硬孔)DEM解析模型可用来反演各孔隙类型的孔隙体积比,它可以通过实验室测量与理论预测之间的平方误差最小反演得到.砂岩样品的反演结果揭示,软孔的孔隙体积百分比与粘土含量没有明显的相关性.     

Physics and Seismic Modeling for Monitoring CO<Subscript>2</Subscript> Storage

We present a new petro-elastical and numerical-simulation methodology to compute synthetic seismograms for reservoirs subject to CO₂ sequestration. The petro-elastical equations model the seismic properties of reservoir rocks saturated with CO₂, methane, oil and brine. The gas properties are obtained from the van der Waals equation and we take into account the absorption of gas by oil and brine, as a function of the in situ pore pressure and temperature. The dry-rock bulk and shear moduli can be obtained either by calibration from real data or by using rock-physics models based on the Hertz-Mindlin and Hashin-Shtrikman theories. Mesoscopic attenuation due to fluids effects is quantified by using White's model of patchy saturation, and the wet-rock velocities are calculated with Gassmann equations by using an effective fluid modulus to describe the velocities predicted by White's model. The simulations are performed with a poro-viscoelastic modeling code based on Biot's theory, where viscoelasticity is described by generalizing the solid/fluid coupling modulus to a relaxation function. Using the pseudo-spectral method, which allows general material variability, a complete and accurate characterization of the reservoir can be obtained. A simulation, that considers the Utsira sand of the North Sea, illustrates the methodology.     

Elasticity of MgO to 130 GPa: Implications for lower mantle mineralogy

The aggregate shear wave velocities of MgO (periclase) have been determined throughout Earth's lower mantle pressure regime approaching 130 GPa using Brillouin spectroscopy in conjunction with synchrotron X-ray diffraction technique in a diamond anvil cell apparatus. We found that the extrapolations of the high-pressure shear wave velocities and shear moduli to ambient pressure are highly consistent with earlier studies. However, the measurements over a wide pressure range revealed that the pressure derivative of the shear modulus (dG/dP = G₀′) of MgO is 1.92(2), which is distinctly lower than that of previous lower-pressure experiments. Compared with the previous results on (Mg,Fe)O ferropericlase, there is no clear correlation between iron content and G₀′. We calculate that the shear wave velocity profile of lower mantle along the adiabatic geotherm applied by the lower G₀′ value of periclase can remarkably well reproduce the global seismological 1-D velocity profile model with uniform composition model. The best-fitting result indicates the possibility of a lower mantle mineralogy with ~ 92 vol.% silicate perovskite phase, implying that the bulk composition of lower mantle is likely not to be pyrolitic but more chondritic. The present acoustic measurements performed over the large pressure range have thus led us to a better understanding of compositional model of the Earth's lower mantle.     

In situ measurements of sound velocities and densities across the orthopyroxene → high-pressure clinopyroxene transition in MgSiO3 at high pressure

Using acoustic measurement interfaced with a large volume multi-anvil apparatus in conjunction with in situ X-radiation techniques, we are able to measure the density and elastic wave velocities (V_P and V_S) for both ortho- and high-pressure clino-MgSiO₃ polymorphs in the same experimental run. The elastic bulk and shear moduli of the unquenchable high-pressure clinoenstatite phase were measured within its stability field for the first time. The measured density contrast associated with the phase transition OEN → HP-CEN is 2.6-2.9% in the pressure of 7-9 GPa, and the corresponding velocity jumps are 3-4% for P waves and 5-6% for S waves. The elastic moduli of the HP-CEN phase are K_S=156.7(8) GPa, G = 98.5(4) GPa and their pressure derivatives are K_S′=5.5(3) and G′ = 1.5(1) at a pressure of 6.5 GPa, room temperature. In addition, we observed anomalous elastic behavior in orthoenstatite at pressure above 9 GPa at room temperature. Both elastic wave velocities exhibited softening between 9 and 13-14 GPa, which we suggest is associated with a transition to a metastable phase intermediate between OEN and HP-CEN.     

Elasticity of (Mg, Fe)(Si, Al)O3-perovskite at high pressure

The most abundant mineral on Earth has a perovskite crystal structure and a chemistry that is dominated by MgSiO₃ with the next most abundant cations probably being aluminum and ferric iron. The dearth of experimental elasticity data for this chemically complex mineral limits our ability to calculate model seismic velocities for the lower mantle. We have calculated the single crystal elastic moduli (c_ij) for (Mg, Fe^3 +)(Si, Al)O₃ perovskite using density functional theory in order to investigate the effect of chemical variations and spin state transitions of the Fe³⁺ ions. Considering the favored coupled substitution of Mg²⁺-Si^4 + by Fe³⁺-Al³⁺, we find that the effect of ferric iron on seismic properties is comparable with the same amount of ferrous iron. Ferric iron lowers the elastic moduli relative to the Al charge-coupled substitution. Substitution of Fe³⁺ for Al³⁺, giving rise to an Fe/Mg ratio of 6%, causes 1.8% lower longitudinal velocity and 2.5% lower shear velocity at ambient pressure and 1.1% lower longitudinal velocity and 1.8% lower shear velocity at 142 GPa. The spin state of the iron for this composition has a relatively small effect (< 0.5% variation) on both bulk modulus and shear modulus.     

Effects of fabric anisotropy on elastic shear modulus of granular soils

The fabric anisotropy of a granular soil deposit can strongly infl uence its engineering properties and behavior. This paper presents the results of a novel experimental study designed to examine the effects of fabric anisotropy on smallstrain stiffness and its evolution with loading on the elastic shear modulus of granular materials under a K0 condition. Two primary categories of fabric anisotropy, i.e., deposition-induced and particle shape-induced, are investigated. Toyoura sand deposits with relative densities of 40% and 80% were prepared using deposition angles oriented at 0o and 90o. Piezoelectric transducers were used to obtain the elastic shear modulus in the vertical and horizontal directions(Gvh and Ghh). The measurements indicate distinct differences in the values of G with respect to the different deposition angles. Particle shapeinduced fabric anisotropy was examined using four selected sands. It was concluded that sphericity is a controlling factor dominating the small-strain stiffness of granular materials. The degree of fabric anisotropy proves to be a good indicatorin the characterization of stress-induced fabric evolution during loading and unloading stress cycles. The experimental data were used to calibrate an existing micromechanical model, which was able to represent the behavior of the granular material and the degree of fabric anisotropy reasonably well.     

Elasticity of MgSiO3 in the ilmenite phase

The single-crystal elastic moduli of the ilmenite phase of MgSiO₃ have been determined from Brillouin spectroscopy. They are: C₁₁ = 472, C₁₂ = 168, C₃₃ = 382, C₁₃ = 70, C₄₄ = 106, C₁₄ = ?27, C₆₆ = 152 and C₂₅ = ?24 in GPa. These elastic properties are consistent with a structural mechanical model where the silicon octahedra are very stiff under compression and shear. This latter property yields an unexpectedly high shear modulus for the magnesium silicate ilmenite as compared with analogue compounds. The further transformation to perovskite will probably be associated with a significant increase in elastic properties since the strong silicon polyhedra form a structural network in this phase. The transformation of spinel and stishovite to ilmenite is associated with a slight density increase and a slight decrease in acoustic velocities. This transformation will probably not produce a seismic discontinuity even if it does occur in the Earth's mantle.     

On Poisson's ratio and composition of the Earth's lower mantle

Poisson's ratio of the lower mantle, calculated from recently published values of seismic wave velocities and extrapolated to atmospheric pressure and room temperature is found to be in the range 0.23 ⩽ ν ⩽ 0.25. These values are compared with the values of Poisson's ratio calculated for binary mixtures of MgSiO₃ perovskite and magnesiowüstite with various iron contents. Current values of the experimental error on measured elastic moduli give little hope to be able to discriminate between pyrolite and chondritic lower mantles: both are acceptable if the shear modulus of perovskite is in the upper range of Liebermann et al. estimates. A similar calculation using the seismic parameter φ confirms the results obtained by considering Poisson's ratio and further constrains the value of the shear modulus of perovskite to lie between 1600 and 1700 kilobars for current mantle models to remain plausible. Chemical stratification of the mantle is, therefore, possible but not required by seismological data.     

Elasticity of polycrystalline stishovite

Ultrasonic data for the velocities of SiO₂-stishovite have been determined as a function of pressure to 10 kbar at room temperature for polycrystalline specimens hot-pressed at pressures P = 120kbar and temperatures T = 900°C. These cylindrical specimens are 2 mm in diameter and 0.9–1.4 mm long and have a grain size less than 10 μm. Compressional and shear wave velocities were measured both parallel and perpendicular to the axis of pressing and were found to be isotropic at 10 kbar with ν_p = 11.0 ± 0.2km/sec andν_s = 6.9 ± 0.3km/sec; this shear velocit is substantially higher than that of Mizutani et al. (1972) perhaps due to the presence of crack orientations in their specimen which affected ν_s but not ν_p. The Murnaghan P-V trajectories calculated from the ultrasonic data [bulk modulus K_s = 2.5 ± 0.3Mbar and assuming (?K_s/?P)_T = 6 ± 2] are consistent with recent hydrostatic compression data and with the shock wave compression data above 600 kbar. The combined evidence from the data of the ultrasonic and hydrostatic compression techniques suggests that the most probable value of the bulk modulus of stishovite at zero pressure is close to the upper limit of the uncertainty of our ultrasonically determined value, K₀ = 2.7?2.8Mbar. Elasticity data for rutile-type oxides are not compatible with normal K_s-V₀ systematics perhaps due to the neglect of non-central forces in the lattice model. These new stishovite data would make it impossible to satisfy the elasticity-density data of the lower mantle using an oxide mixture with either olivine or pyroxene stoichiometry.     

Recovering Shear-wave Anisotropy of the Lower Crust: The Influence of Systematic Errors on Travel-time Inversion

—Studies of seismic anisotropy in situ can help to discriminate between different rock types for the lower crust. In this context we investigate the sensitivity of an iterative linearized 3-D travel-time inversion scheme for transversely isotropic media with respect to two types of systematic errors wrong velocities and interface topography of the hanging wall of the lower crust. The computations simulate realistic field conditions such as found for the Variscan crust at the Urach geothermal anomaly. The study focusses on the possible information content of split S _M ?S arrivals observed along two orthogonal expanding spread profiles. It ensues that an imperfect knowledge of the layer geometry is of minor importance compared to errors in the velocities of the hanging wall. In particular, upper crust anisotropy has to be considered carefully. Generally, the anisotropy of transversely polarized shear waves (SH waves) was recovered with higher accuracy than the anisotropy of vertically polarized shear waves (SV waves).     

Rock physics modelling of porous rocks with multiple pore types: a multiple-porosity variable critical porosity model

A critical porosity model establishes the empirical relationship between a grain matrix and a dry rock by the concept of critical porosity. The model is simple and practical and widely used. But the critical porosity in the model is a fixed value that cannot relate to pore structure. The aim of this paper is to establish the theoretical relationship between critical porosity and pore structure by combining Kuster–Toksöz theory with the critical porosity model. Different from the traditional critical porosity model, critical porosity is not an empirical value but varied with pore shape and the ratio of bulk modulus versus shear modulus of the grain matrix. The substitution of the theoretical relationship into Kuster–Toksöz theory will generate the formulae for the bulk and shear moduli of multiple-porosity dry rocks, which is named the multiple-porosity variable critical porosity model. The new model has been used to predict elastic moduli for sandstone and carbonate rock. We compare the modelling results for P- and S-wave velocities and elastic moduli with the experimental data. The comparison shows that the new model can be used to describe the elastic properties for the rocks with multiple pore types.     

Review of correlations between SPT N and shear modulus: A new correlation applicable to any region

A low strain shear modulus plays a fundamental role in earthquake geotechnical engineering to estimate the ground response parameters for seismic microzonation. A large number of site response studies are being carried out using the standard penetration test (SPT) data, considering the existing correlation between SPT N values and shear modulus. The purpose of this paper is to review the available empirical correlations between shear modulus and SPT N values and to generate a new correlation by combining the new data obtained by the author and the old available data. The review shows that only few authors have used measured density and shear wave velocity to estimate shear modulus, which were related to the SPT N values. Others have assumed a constant density for all the shear wave velocities to estimate the shear modulus. Many authors used the SPT N values of less than 1 and more than 100 to generate the correlation by extrapolation or assumption, but practically these N values have limited applications, as measuring of the SPT N values of less than 1 is not possible and more than 100 is not carried out. Most of the existing correlations were developed based on the studies carried out in Japan, where N values are measured with a hammer energy of 78%, which may not be directly applicable for other regions because of the variation in SPT hammer energy. A new correlation has been generated using the measured values in Japan and in India by eliminating the assumed and extrapolated data. This correlation has higher regression coefficient and lower standard error. Finally modification factors are suggested for other regions, where the hammer energy is different from 78%.     

Relations of electrical conductivity to physical conditions within the asthenosphere

This paper examines the constraint placed by electrical conductivity on physical conditions existing within the mantle. Determination of bulk electrical conductivities of multiphase materials from knowledge of individual phase conductivities and their relative fractions is discussed in the light of recent studies of the equilibrium phase geometries of partial melts. It is concluded that existing models which are based on assumed geometries offer little refinement over Hashin-Shtrikman variational bounds. The possible effects of the gravitational field on phase geometry in partial melts and the resulting conductivity anisotropy are considered. At this time it appears that minimum melt fractions can be safely estimated from a knowledge of Earth conductivity combined with laboratory data on melt and crystalline conductivities and the Hashin-Shtrikman upper bound. The question of whether or not the asthenosphere corresponds to a zone of partial melting is also addressed.     

Longitudinal vibration of non-homogeneous earth dams

Two-dimensional analytical elastic models are developed for evaluating dynamic characteristics, namely natural frequencies and modes of vibration of a wide class of earth dams in a direction parallel to the dam axis. In these models the non-homogeneity of the dam materials is taken into account by assuming a specific variation of the stiffness properties along the depth (due to the continuous increase in confining pressure). In addition, both shear and normal (axial) deformations are considered. Cases having constant elastic moduli, linear and trapezoidal variations of elastic moduli, and elastic moduli increasing as the one-half, one-third, two fifths, and a general (l/m)th powers of the depth are studied. Dynamic properties of three real earth dams in a seismically active area (Southern California) estimated from their earthquake records (input ground motion and crest response in the longitudinal direction) as well as results from full-scale dynamic tests on one of these dams (including ambient and forced vibration tests) are compared with those from the suggested models. It was found that the models in which the shear modulus and the modulus of elasticity of the dam material vary along the depth are the most appropriate representations for predicting the dynamic characteristics. The agreement between the experimental and earthquake data and the theoretical results from some of the models is reasonably good.     

A semi-empirical velocity-porosity-clay model for petrophysical interpretation of P- and S-velocities

We design a velocity–porosity model for sand-shale environments with the emphasis on its application to petrophysical interpretation of compressional and shear velocities. In order to achieve this objective, we extend the velocity–porosity model proposed by Krief et al., to account for the effect of clay content in sandstones, using the published laboratory experiments on rocks and well log data in a wide range of porosities and clay contents. The model of Krief et al. works well for clean compacted rocks. It assumes that compressional and shear velocities in a porous fluid-saturated rock obey Gassmann formulae with the Biot compliance coefficient. In order to use this model for clay-rich rocks, we assume that the bulk and shear moduli of the grain material, and the dependence of the compliance on porosity, are functions of the clay content. Statistical analysis of published laboratory data shows that the moduli of the matrix grain material are best defined by low Hashin–Shtrikman bounds. The parameters of the model include the bulk and shear moduli of the sand and clay mineral components as well as coefficients which define the dependence of the bulk and shear compliance on porosity and clay content. The constants of the model are determined by a multivariate non-linear regression fit for P- and S-velocities as functions of porosity and clay content using the data acquired in the area of interest. In order to demonstrate the potential application of the proposed model to petrophysical interpretation, we design an inversion procedure, which allows us to estimate porosity, saturation and/or clay content from compressional and shear velocities. Testing of the model on laboratory data and a set of well logs from Carnarvon Basin, Australia, shows good agreement between predictions and measurements. This simple velocity-porosity-clay semi-empirical model could be used for more reliable petrophysical interpretation of compressional and shear velocities obtained from well logs or surface seismic data.     

Elasticity of single-crystal SmAlO3, GdAlO3 and ScAlO3 perovskites

The adiabatic single-crystal elastic moduli of SmAlO₃, GdAlO₃ and ScAlO₃, all with the orthorhombic perovskite structure, have been measured by Brillouin spectroscopy under ambient conditions. These 3 compounds display various degrees of crystallographic distortion from the ideal cubic perovskite structure. We find that longitudinal moduli in directions parallel to the axes of a pseudocubic subcell are nearly equal and insensitive to distortions of the crystal structure from cubic symmetry, whereas, the moduli C₁₁ and C₂₂, parallel to the orthorhombic axes, display pronounced anisotropy with the exception of ScAlO₃. The shear moduli also correlate with distortion from cubic symmetry, as measured by rotation, or tilt angles, of the AlO₆ octahedra. Our data support the observations of Liebermann et al. that perovskite-structure compounds define consistent elasticity trends relating bulk modulus and molar volume, and sound speed and mean atomic weight. These relationships have been used to estimate bulk and shear moduli for the high-pressure polymorphs of CaSiO₃ and MgSiO₃ with the perovskite structure.     

Static and dynamic elastic moduli of organic-rich chalk

The elastic moduli and anisotropy of organic-rich rocks are of great importance to geoengineering and geoprospecting of oil and gas reservoirs. In this paper, we probe into the static and dynamic moduli of the Ghareb–Mishash chalk through laboratory measurements and new analytical approaches. We define a new anisotropy parameter, ‘hydrostatic strain ratio’ (Ω), which describes the differential contraction of anisotropic rocks consequent to hydrostatic compression. Ω depends on the C₁₁, C₁₂, C₁₃ and C₃₃ stiffness constants of a transversely isotropic material, and therefore enables a unique insight into the anisotropic behaviour of TI rocks. Ω proves more sensitive to anisotropy within the weak anisotropy range, when compared with Thomsen's ε and γ parameters. We use Ω to derive static moduli from triaxial compression tests performed on a single specimen. This is done by novel employment of a hydrostatic-deviatoric combination for transversely isotropic elastic stiffnesses. Dynamic moduli are obtained from acoustic velocities measurements. We find that the bedding-normal velocities are described well by defining kerogen as the load-supporting matrix in a Hashin–Shtrikman model (‘Hashin–Shtrikman (HS) kerogen’). The dynamic moduli of the Ghareb–Mishash chalk in dry conditions are significantly higher than the static moduli. The dynamic/static moduli ratio decreases from ∼4 to ∼2 with increasing kerogen content. Both the static and dynamic moduli decrease significantly with increasing porosity and kerogen content. The effect of porosity on them is two times stronger than the effect of kerogen.     

Effect of Time on Dynamic Shear Modulus of Selected Cohesive Soil of One Section of Express Way No. S2 in Warsaw

Several researches published comprehensive reports on dynamic soil properties of cohesive soils, in which many of them outlined, i.e., key factors affecting the dynamic shear modulus. For cohesive soils, the modulus at small strains (g < 10^t-3 %) is, first of all, a function of void ratio and effective confining stress. For clays, however, secondary time effects and clay mineralogy (fabric and structure) also appear to be important. The influence of confinement of laboratory-prepared as well as naturally deposited clays consists in an increase of shear modulus logarithmically as a function of time. In this paper, the effect of duration of the various confining pressures on dynamic shear modulus (G) of selected cohesive soils from Warsaw area was evaluated. Shear modulus was determined on the basis of resonant column tests, at low and high shearing strain amplitudes. It is shown that the calculated shear modulus is time-dependent; during approximately first 1000 minutes of consolidation, the moduli increased by almost 50%. Moreover, it is characterized by two phases: an initial one results from primary consolidation and a second one, which occurs after the end of primary consolidation, herein about 16-17 hours, and is called “long-term time effect”. This effect was found also for modulus at higher shearing strains (γ > 10³%, e.g., 3 × 10³%, 5 × 10³%, 8 × 10³%, 2 × 10³%).     

Spatial relation between source regions of strong earthquakes (M ≥ 7) on Southern Kamchatka and the distribution of velocities and elastic moduli in the Benioff zone

The presence of a phenomenological relationship between high velocity regions in the Benioff zone and sources of relatively strong earthquakes (M ≥ 6) was established for the first time from the comparison of such earthquakes with the velocity structure of central Kamchatka in the early 1970s. It was found that, in the region with P wave velocities of 8.1–8.5 km/s, the number of M ≥ 6 earthquakes over 1926–1965 was 2.5 times greater than their number in the region with velocities of 7.5–8.0 km/s. Later (in 1979), within the southern Kurile area, Sakhalin seismologists established that regions with V _P = 7.3–7.7 km/s are associated with source zones of M = 7.0–7.6 earthquakes and regions with V _P = 8.1–8.4 km/s are associated with M = 7.9–8.4 earthquakes. In light of these facts, we compared the positions of M = 7.0–7.4 earthquake sources in the Benioff zone of southern Kamchatka over the period 1907–1993 with the distribution of regions of high P velocities (8.0–8.5 to 8.5–9.0 km/s) derived from the interpretation of arrival time residuals at the Shipunskii station from numerous weak earthquakes in this zone (more than 2200 events of M = 2.3–4.9) over the period 1983–1995. This comparison is possible only in the case of long-term stability of the velocity field within the Benioff zone. This stability is confirmed by the relationship between velocity parameters and tectonics in the southern part of the Kurile arc, where island blocks are confined to high velocity regions in the Benioff zone and the straits between islands are confined to low velocity regions. The sources of southern Kamchatka earthquakes with M = 7.0–7.4, which are not the strongest events, are located predominantly within high velocity regions and at their boundaries with low velocity regions; i.e., the tendency previously established for the strongest earthquakes of the southern Kuriles and central Kamchatka is confirmed. However, to demonstrate more definitely their association with regions of high P wave velocities, a larger statistics of such earthquakes is required. On the basis of a direct correlation between P wave velocities and densities, the distributions of density, bulk modulus K, and shear modulus μ in the upper mantle of the Benioff zone of southern Kamchatka are obtained for the first time. Estimated densities vary from 3.6–3.9 g/cm³ in regions of high V _P values to 3.0–3.2 g/cm³ for regions of low V _P values. The bulk modulus K in the same velocity regions varies from (1.4–1.8) × 10¹² to (0.8–1.1) × 10¹² dyn/cm², respectively, and the shear modulus μ varies from (0.8–1.0) × 10¹² to (0.5–0.7) × 10¹² dyn/cm², respectively. Examination of the spatial correlation of the source areas of southern Kamchatka M = 7.0–7.4 earthquakes with the distribution of elastic moduli in the Benioff zone failed to reveal any relationship between their magnitudes and the moduli because of the insufficient statistics of the earthquakes used.