Compressional Wave Velocity Research Articles

Machine learning models to estimate stress wave velocities of cohesionless soils during triaxial compression influenced by particle characteristics

Stress wave velocities, i.e. shear and compression wave velocities and hence small strain stiffness, are important parameters required for dynamic analysis and designing various geotechnical structures. Measuring wave velocities has been challenging, let alone developing a model for their prediction. There exist few models in the literature; however, their use is limited and can be used mostly in specific cases, for example, isotropic conditions. This paper uses the dataset developed in the author's previous study and aims to develop a model where the wave velocities can be predicted accurately. The dataset contains wave velocities evolution of three different materials: Toyoura sand, river sand and silica sand at three different initial relative densities sheared by increasing the major principal stress. The three different materials allowed the study of different particle characteristics, including sphericity, convexity, aspect ratio and roughness, in predicting and modelling the wave velocities. This paper utilizes three different machine learning (ML) models viz. artificial neural network (ANN), support vector regression (SVR) and gradient boosting regression (GBR) to model the observed behaviour of wave velocities evolution. All three models were equally efficient in modelling the evolution of both shear (Vs) and compression wave (Vp) velocities. The predicted Vs and Vp using the three different machine learning models are also compared with a literature model available for isotropic stress conditions and showed that the ML models developed in this study are superior by incorporating the effect of grain characteristics which is missing from the literature model. The relationship between the prediction of the wave velocities and input parameters is highlighted using feature importance and Pearson's coefficient. At the end of this study, ANN is used to provide a set of nonlinear equations in the matrix form for Vs and Vp, which can be used by anyone and does not require any prior knowledge of ML or coding.

Estimating Effective Material Parameters of Inhomogeneous Layers Using Finite Element Method

This study presents a finite element method (FEM) approach to estimate the effective medium parameters of 2-D and 3-D layers of arbitrary composition. The resonance frequency of a layer to be investigated is found by exciting the layer with plane waves and studying the reflected sound pressure from the layer as a function of frequency and incidence angle. This allowed for the calculation of compressional and shear wave velocities. The method was validated by applying the method to layers with known acoustic parameters and by comparing with results from the established analytical models. Composite layers with 1-3 and 2-2 connectivity are well described by established effective-medium theories, but these require the composite structures to be small compared to the acoustic wavelength. This limitation was overcome by the described FEM-based model, which could also capture deviations occurring in coarser composites. Conventional analytical models predict wave velocities as a function of void concentration, not considering positions of the voids. The described FEM approach predicted up to 5% variation in wave velocities for gold layers with identical volume fraction of voids, depending on the void distribution. This demonstrates that void positions influence wave velocity. The influence of connectivity between inclusions was studied by modeling tungsten inclusions in an epoxy matrix. It was found that composites with inclusions connected in a preferred direction had higher wave velocity in the direction of connectivity compared to randomly oriented inclusions. It is concluded that the presented FEM model reproduces the literature values for homogeneous materials and agrees with effective medium theories for fine-pitched composites. However, the strength of the model is its ability to go beyond this and model phenomena in real finite-size composites not captured by the classic effective medium models.

Normal mode observability of radial anisotropy in the Earth’s mantle

SUMMARYObservations of seismic anisotropy provide useful information to infer directions of mantle flow. However, existing global anisotropic tomography models are not consistent, particularly in the lower mantle. Therefore, the interpretation of seismic anisotropy in terms of mantle dynamics and evolution remains difficult. While surface and body waves are commonly used to build radially anisotropic tomography models, they provide heterogeneous data coverage and the radial anisotropy structure retrieved using these data may be biased by the use of imperfect crustal corrections. Normal modes, the free oscillations of the Earth, automatically provide global data coverage and their sensitivity to shear wave (vs) and compressional wave (vp) velocity makes them suitable to study both vs and vp anisotropy in the mantle. In this study, we assess whether current normal mode splitting data have sufficient sensitivity to lower mantle anisotropy to potentially constrain it. We consider the uncertainties in the data and the effect of inaccuracies in crustal thickness corrections and the assumed scaling between vp and vs. We perform forward modelling of normal mode data using six different 3-D global radially anisotropic tomography models to document how strong and widespread anisotropy has to be to be observable in current normal mode data. We find that, on average 50% of the spheroidal and 55% of the toroidal modes investigated show significant sensitivity to vs anisotropy, while roughly 57% of the spheroidal modes also have strong sensitivity to vp anisotropy. Moreover, we find that the normal mode data fit varies substantially for the various anisotropic tomography models considered, with the addition of anisotropy not always improving the data fit. While we find that crustal thickness corrections do not strongly impact modes that are sensitive to the lower mantle, we observe a trade-off between radial anisotropy and vp scaling for these modes. As long as this is taken into consideration, our findings suggest that existing normal mode data sets can provide valuable information on both vs and vp anisotropy in the mantle.

Rock physics modeling with mineralogical inversion in sandstone reservoir

Reservoir characterization is a valuable tool for the oil and gas industry. A better understanding of the reservoir demands the integration of petrophysical properties and elastic parameters. This integration is commonly performed with the aid of rock physics models. The elastic properties of solid-phase components are important parameters for rock physics models calibration. In this paper, we use an adaptation of mineralogical inversion to estimate the volumes of quartz, feldspar, and clay minerals that incorporate previously calculated clay volume and porosity. Clay volume estimation is performed from neutron and bulk density logs. We chose this method due to the presence of feldspar minerals in the reservoirs. Our workflow consists of petrophysics properties estimation, mineralogical inversion, rock physics model calibration in Well A, and compressional-wave velocity estimation in Well B. The mineralogical inversion in Well A provided average volumes of 64% quartz and 16% feldspar and previously estimated 20% clay minerals. When applied to Well B, the calibrated soft-sand model is a better approximation for high porosity data points than the constant-cement model, and the error between original and estimated logs is about 2.9%, suggesting that the approach can be extended to other wells in the study area.

Sound velocity of hexagonal close-packed iron to the Earth’s inner core pressure

Here we determine the compressional and shear wave velocities (vp and vs) of hexagonal close-packed iron, a candidate for the main constituent of the Earth’s inner core, to pressures above 300 gigapascals using a newly designed diamond anvil cell and inelastic X-ray scattering combined with X-ray diffraction. The present results reveal that the vp and vs of the Preliminary reference Earth model (PREM) inner core are 4(±2)% and 36(±17)% slower than those of the pure iron, respectively at the centre of the core. The density and sound velocity of the PREM inner core can be explained by addition of 3(±1) wt% silicon and 3(±2) wt% sulphur to iron‒5 wt% nickel alloy. Our suggested inner core composition is consistent with the existing outer core model with oxygen, as the growth of the inner core may have created a secular enrichment of the element in the outer core.

In-situ Primary Stress Detection Based on Seismic Tomography Measurements and Numerical Back-analysis for an Underground Radwaste Repository

In order to understand how the in-situ primary stress state has evolved with subsidence and uplift in a granitic rock mass for anticipated of a radioactive waste repository in Hungary, the authors investigated the applicability of seismic tomography as an interpretive tool. Very high P wave velocity (Vp) values were obtained during the tomographic scanning of the study area of the repository, and these were compared with existing findings of in-situ and laboratory seismic measurements.Apart from seismic tomographic survey, dynamic FEM numerical modelling, empirical calculations of residual stresses, laboratory measurements of compression wave (ultrasonic) velocities on intact rock cores, in-situ primary stress measurements as well as site geological model were integrated to evaluate the use of seismic tomography for identifying possible in-situ stress increases around the excavation.A detailed calibration modelling was carried out based on the site seismic tomography measurements and during the large-scale modelling. It was observed that the increasing Vp is directly related to simulated increasing directional loadings on the rock mass. Using a measured wave raypath it was possible to check the different in-situ stress parametrizations which resulted in the best approximation to the measured Vp values.It was concluded that the rock mass under investigation to extend the repository must have higher in-situ stress values than the area of the constructed deposition chambers nearby. The results of this research indicated that seismic tomography is a useful tool for determining relative stress around and within the vicinity of underground excavation.

Permeability and Rigidity of Green River Shale Before and After Exposure to Water

Abstract Acoustic travel times through Green River shale samples both parallel and perpendicular to bedding are measured to investigate water interactions with Green River shale and the impact of pertinent factors including exposure durations, pore pressure, effective stress, and anisotropy. To assist these analyses, X-ray diffraction (XRD) and permeability are also measured. Understanding petrophysical and rock properties before and after exposure to water is essential to optimize stimulation design in shale reservoirs. The XRD shows that the samples are clay-poor and mainly consist of carbonate minerals. Bedding-parallel and bedding-perpendicular permeability to non-sorbing He are measured before exposure to tap water to analyze permeability evolution under different stress conditions using the pressure transient method. The samples record very low permeabilities at recreated confining stresses, indicating that the shale requires stimulation. The permeability decreases as confining stress increases, while the permeability increases with increasing pore pressure. Acoustic travel time measurements show that compressional and shear wave velocities, and elastic moduli of the samples increase as confining stress increases before samples are exposed to tap water. This results in more rigid samples which exhibit higher fracture conductivity. The samples are saturated with tap water in containers at atmospheric pressure after the initial acoustic travel time measurements. Further acoustic travel time measurements and chemical elemental analysis in tap water show that quartz dissolves from the pore structure of the Green River shale, reducing the acoustic velocities, and elastic moduli of the samples that then exhibit lower rigidity and fracture conductivity.

Prediction of compressional wave velocity of cement-reinforced soil from core images using a convolutional neural network regression model

This study aims to predict the compressional wave velocity (Vp) from the photographic images of cylindrically cored cement-reinforced soils using a convolutional neural network (CNN) model. The experimentally measured Vp values and corresponding surficial core images were subjected to the CNN regression model based on a backbone network pre-trained by transfer learning. The model was retrained by fine-tuning and optimized with regularization strategies and data augmentation. The results showed that the trained network model reliably predicted Vp with reasonable performance of R2 = 0.78. Three-dimensional X-ray computed tomographic imaging of both overestimated and underestimated specimens revealed that surficial core images did not sufficiently reflect the internal structures. The slightly scattered prediction seemed attributed to the insufficient dataset size and invisible internal structure. Nevertheless, the proposed approach allowed not only estimating Vp at unmeasured spots in cores based on core images by fully leveraging artificial intelligence but also obtaining consecutive Vp profiles. Furthermore, this study established the hardly seen correlation between core image and Vp by the proposed CNN regression model and can be extended to estimation of other geophysical and geomechanical properties to construct a sufficient dataset for subsurface geostatistical modeling.

A New Seismic Inversion Scheme Using Fluid Dispersion Attribute for Direct Gas Identification in Tight Sandstone Reservoirs

Sufficient gas accumulation is an essential factor that controls the effective development of tight sandstone gas reservoirs that are generally characterized by low porosity and permeability. Seismic methods are important for predicting potential gas areas in tight sandstones. However, the complex relationships between rock physical properties and gas saturation make gas enrichment estimation with seismic methods challenging. Nonetheless, seismic velocity dispersion using a wave-induced fluid flow mechanism can enable gas identification by utilizing the associated dispersion attributes. This paper proposes a method for improved gas identification using a new fluid dispersion attribute obtained by incorporating the decoupled fluid-solid seismic amplitude variation with offset representation into the frequency-dependent inversion scheme. Numerical analyses and synthetic data tests confirmed the enhanced sensitivity of the fluid dispersion attribute to gas saturation compared to the conventionally used compressional wave velocity dispersion attribute. Field data applications further validated the ability of the proposed fluid dispersion attribute to improve gas prediction in tight sandstone reservoirs. The results of the measurements enable rational interpretation of the geological significance of assessments of reservoir properties from gas-producing wellbores. The proposed fluid dispersion attribute is a reliable indicator for gas prediction and represents a useful tool for characterizing tight sandstone reservoirs.

Investigating the elastic, mechanical, and thermal properties of polycrystalline Mo2C under high pressure and high temperature

Understanding pressure-induced acoustic velocity-elasticity behavior has extraordinary significance in the fields of materials science and earth science. Herein, we used an advanced high-pressure high-temperature (HPHT) method to synthesize pure-phase β-Mo2C bulk ceramics. The sound velocity-elasticity behavior, thermodynamic properties, and mechanical behavior of the synthesized β-Mo2C ceramics were systematically investigated by in-situ high-pressure ultrasonic interferometry and theoretical calculations. The compressive and shear wave velocities, isothermal bulk modulus, shear modulus, Vickers hardness, fracture toughness, Debye temperature, and melting temperature of polycrystalline β-Mo2C exhibited a monotonic increase with increasing pressure. The experimental and calculation results showed that β-Mo2C had strong ductile behavior and was a ductile ceramic. Additionally, the temperature-hardness relationship of the as-synthesized polycrystalline β-Mo2C was investigated by in-situ high-temperature Vickers indentation measurements. The hardness of β-Mo2C gradually decreased with increasing temperature, and this ceramic still maintained a hardness as high as 12.3 GPa at 500 °C. These results suggest that the intrinsic mechanical and thermodynamic properties of β-Mo2C are dominated by its unique electronic structure and bonding mode.

Non-Destructive Evaluation of Mortar with Ground Granulated Blast Furnace Slag Blended Cement Using Ultrasonic Pulse Velocity.

Non-destructive evaluation using ultrasonic pulse velocity (Vp) testing has extensive applications in the concrete industry. With advances in construction technology, the use of ground granulated blast furnace slag (GGBFS) as a partial replacement to cement in a concrete mix is growing in popularity primarily because it reduces the initial capital cost of raw materials and the associated energy costs. This paper investigates the effect of the water-to-cement (wc) ratio and the cement content replaced by GGBFS on the development with time of the ultimate compressive strength () and the compression wave velocity (Vp) of mortar. The results showed that in the case of mortar with higher percentages of GGBFS replacement (where nucleation surfaces are more abundant), increasing wc can increase but cause a decrease in Vp. The posterior hydration process is highly dependent upon the water particles in the mixture after the first stage of hydration. After 7 days of curing, experimental results show that the of slag blended cement mix design wc ratio of 0.6 surpassed the value of an Ordinary Portland cement. A regression model correlating the and Vp of slag blended mortar is developed, which can be used to predict at concrete ages ranging from 1 day to 28 days for mixes with GGBFS percentage replacement values ranging from 15% to 45%.

Understanding carbonates from Saudi Arabia: Laboratory measurements of elastic properties. Part I

Laboratory measurements of elastic properties are presented for carbonate samples from the Saudi Arabian Arab formation. Compressional and shear wave velocities (at 1 MHz) and the quasi-static strains of 37 carbonate rock samples were measured as functions of the saturating fluid and the confining pressure. Furthermore, P- and S-wave velocities of the saturated samples were measured at constant differential pressure of 15 MPa. The lithology, mineralogy, porosity, pore type and pore size distribution of each sample were obtained using a combination of thin section and scanning electron microscopy, helium porosimetry and mercury intrusion porosimetry. The samples showed a high sensitivity to the applied differential pressure especially for the high porosity samples. Due to the lack of closing microcracks and compliant pores in low porosity samples, the travel times show slight changes with the confining pressure whereas the high porosity samples showed a remarkable reduction of the travel time with an increase in confining pressure for both P- and S-waves. We found that the sample physically deformed at pressure above 25 MPa. Evidence of inelastic deformation were observed in a few samples even at a differential pressure of 25 MPa. Both P- and S-wave velocities show a visible deviation with respect to the conventional velocity-porosity inverse relationships, with large variabilities in the velocities at a given porosity. The tight samples only measured a 3% average increase in velocity over the pressure range, whereas the porous samples measured a 20% average increase in velocity over the same pressure range. Despite the nature of porosity, the measured velocities showed a general increase with pressure. Also, all measured samples showed an increase in the bulk modulus of the rock after water saturation with 60% and less than 1% increases. There was no trivial correlation between porosity and the magnitude of the bulk modulus increase. Indeed, the shear modulus of the rock should be insensitive to any changes in water saturation, however, we observed some changes. The presence of anhydrite reduced both P- and S-wave velocities and had a significant influence on the velocities of carbonate rocks at any given porosity. A second paper will build on the results and will consist of predicting and validating suitable rock physics models using the measured carbonate data. The data that support findings of this study are openly available in ERA: Education and Research Archive University of Alberta at https://doi.org/10.7939/R3091B

Sound velocity anomalies of limestone at high pressure and implications for the mantle wedge

ABSTRACT Limestone mainly consisting of CaCO3 is one of the most abundant carbonates on Earth's surface. The sound velocities of limestone at high pressure were determined at room temperature using ultrasonic interferometry in a multi-anvil apparatus. Softening and discontinuities in compressional (P) and shear (S) wave velocities have been observed at around 1.45 GPa due to the phase transition of CaCO3-I to CaCO3-II, and the coexistence of CaCO3-IIIb and CaCO3-III stay up to ∼5 GPa. Limestone under CaCO3-I and CaCO3-III phase has much lower velocities and higher V P/V S ratio than most crustal and mantle minerals and the PREM model. Phase transitions from CaCO3-I to CaCO3-II cause the abrupt reduction of V P, V S, and V P/V S. The low velocities and high/low V P/V S ratio are well consistent with the seismology observed in the mantle wedge. The result suggests that the subduction of limestone into Earth's interior would cause low-velocity anomalies in mantle wedges.

Characterizing Gas Hydrate–Bearing Marine Sediments Using Elastic Properties—Part 1: Rock Physical Modeling and Inversion from Well Logs

Gas hydrates are considered a potential energy source for the future. Rock physics modeling provides insights into the elastic response of sediments containing gas hydrates, which is essential for identifying gas hydrates using well-log data and seismic attributes. This paper establishes a rock physics model (RPM) by employing effective medium theories to quantify the elastic properties of sediments containing gas hydrates. Specifically, the proposed RPM introduces critical gas hydrate saturation for various modeling schemes. Such a key factor considers the impact of gas hydrates on sediment stiffnesses during the dynamic process of the gas hydrate accumulating as pore fillings and part of the solid components. Theoretical modeling illustrates that elastic characteristics of the sediments exhibit distinct variation trends determined by critical gas hydrate saturation. Numerical tests of the model based on the well-log data confirm that the proposed technique can be employed to rationally predict gas hydrate saturation using the elastic properties. The compressional wave velocity model is also developed to estimate the gas hydrate saturation, which gives reliable fit results to core measurement data. The proposed methods could improve our understanding of the elastic behaviors of gas hydrates, providing a practical approach to estimating their concentrations.

Rock characterization integrating free-free resonant column measurements and X-ray computed tomography imaging

This study addresses the innovative integration of free-free resonant column measurements (FFRC) of body wave velocities in rock specimens with the X-ray microcomputer tomography (micro-CT or μ-CT) imaging. FFRC seismic measurements combined with μ-CT scanning is a novelty manner to execute and interpret dynamic rock core characterization in laboratory. FFRC measurements covers unconstrained compression and shear wave velocities (Vc and Vs) as well as quality factors (Qc and Qs), which were evaluated in the ranges of frequency from ~ 3 to ~ 10 kHz for Berea sandstone specimens and from ~ 10 to ~ 20 kHz for Indiana limestone samples. In addition, the propagation velocity of constrained compression waves, Vp, were measured for all the specimens using direct-travel times. The μ-CT scanning provided micro-structural rock parameters such as porosity, profiles of coefficients of attenuation, as well as pore-size and grain-size distributions. Also, resonance testing allowed us to evaluate the integrity of the rock cores, which could be independently confirmed by using μ-CT scanning along the specimens. Finally, FFRC combined with μ-CT might provide quantitative information to clarify atypical seismic results of rock cores.

Compressional and shear wave velocities relationship in anisotropic organic shales

Understanding the relationship between compressional and shear wave velocities and their anisotropic characteristics in self-resourcing reservoir shales is of considerable interest for petrophysical, geophysical, and geomechanical applications of unconventional resources exploration and production. We compile laboratory measurements on the ultrasonic velocity of known shale reservoirs worldwide. Their compressional and shear wave velocities propagating in the vertical and horizontal directions can be characterized by a linear model with the coefficients of determination (R2) close to 0.9. The P-to-S-wave velocity ratio in the vertical direction is overall higher than that in the horizontal direction. The P-to-S-wave velocity ratio decreases with decreasing P-wave velocity due to the increasing organic matter content and hydrocarbon-filled porosity. Mineralogy influences the relationship between compressional and shear wave velocities to a minor degree. Carbonate-rich shales have a higher P-to-S-wave velocity ratio than silica-rich shale when the P-wave velocity is lower than 4 km/s and a lower ratio when the P-wave velocity is higher than 5 km/s. By employing anisotropic fluid substitution, we justify that the model derived from the laboratory measurements can apply to shale reservoirs containing gas or volatile oil. We apply the linear models to predict the shear wave velocity in two vertical wells of shale reservoirs. The predictions of shear wave velocity in the targeted reservoirs have less than 3% errors, demonstrating that the proposed linear models, to the first order, can predict shear wave velocity in unconventional shale reservoirs.

First-principles study of water incorporation in Fe-containing wadsleyite

Using first-principles density functional theory (DFT), we study the problem of water incorporation in iron bearing wadsleyite ((Mg,Fe)2SiO4) at transition zone pressures and temperatures under varying conditions of vacancy and ferric ion concentration. Our calculations suggest that about 30% of the Fe3+ may be found at Si tetrahedral site, which is in accordance with the experimental report of Bolfan-Casanova et al. [2012]. Considering first-principles optimized crystal structures of hydrous Fe bearing wadsleyite, we calculate their elastic properties, such as bulk and shear moduli, compressional and shear wave velocity, as a function of pressure and temperature. Our calculations find that pressure suppresses the P and S wave ‘velocity-reduction’ that is caused by hydration. However, the suppression is not significant. So it is very unlikely that the pressure suppression will severely affect the accuracy of seismic waves in detecting the water content of the mantle transition zone as recently proposed [Buchen et al., 2018]. Comparison of our theoretical results with the existing seismic data indicates that the water content of the transition zone decreases with increasing depth, with the region close to the 410 km discontinuity being saturated in water, corroborating the previously proposed theories on mantle convection [Bercovici and Karato, 2003].

Influencing Factors of Shear Wave Radiation of a Dipole Source in a Fluid-Filled Borehole

In shear wave far detection logging, dipole-source radiation is the main factor influencing the amplitude of the reflected shear waves. In this paper, a method is derived with the far-field asymptotic solution to calculate the dipole-source radiation of shear waves in a fluid-filled borehole. Then the dipole-source radiation of the shear waves is simulated under both low and high frequencies. In addition, the influences of formation elastic parameters on the dipole-source radiation of the shear waves are analyzed and the variations of the radiation characteristics of the shear wave with source main frequency and borehole radius are compared. Results show that the density and compressional wave velocity of the formation have little effect on the dipole-source radiation of the shear waves. However, the shear wave velocity not only affects the shear wave amplitude radiated to the formation by the dipole source (radiation performance), but also affects the energy distribution of the shear wave at different locations in space (radiation direction). The dipole source has better radiation performance and radiation coverage at low frequency and the optimal excitation frequency in different formations is very close, which is good for the application of this technology under different circumstances. At low frequency, the borehole has little influence on the dipole-source radiation, no matter how large the borehole radius is. However, at high frequency, the borehole modulation of the dipole-source radiation cannot be ignored, especially at large borehole radius.

Experimental Study of Cement Mortar on Compressive Strength and Wave Velocity Characteristics for Different Sand/Cement Ratio and Particle Sizes

Cement mortar is the most commonly used cement-based material, which widely used in construction materials, filling materials and other fields. The aim of this paper is to study the compressive strength and wave velocity characteristic in different particle sizes and sand/cement ratio. The uniaxial compressive strength test and wave velocity test were carried out in different particle size and sand content of mortar samples, respectively. It was found that the compressive strength and wave velocity decrease with the increase of sand/cement ratio. It is interesting that compressive strength and wave velocity in all specimens gradually increases with the increase of particles sizes. The compressive strength of cement mortar is related to the wave velocity. In detailed, the compressive strength and wave velocity was fit to linear relationship. It could thus be used to predict the compressive strength of cement mortar by wave velocity.

Uncertainty quantification for characterization of rock elastic modulus based on P-velocity

ABSTRACT The elastic modulus of rock is an important parameter in rock engineering, but the common methods based on laboratory tests are laborious, especially for obtaining the probability distribution of the elastic modulus that is required in reliability-based design. Many scholars have studied the regression model between the elastic modulus and P-wave velocity; however, most previous reports have ignored the characterization of parameter variability and model uncertainty. To address this problem, a large number of granite samples are collected from the Yingliangbao hydropower station (YLB), compressive wave velocity (P-wave velocity) and uniaxial compression tests are carried out in the laboratory. Then, four different regression models based on the frequentist method and Bayesian method are established to estimate the elastic modulus, the normal priors are adopted by prior analysis and the widely applicable information criterion (WAIC) is used to select the most appropriate Bayesian regression model. Finally, the effects of sample size and sample selection on different methods are studied, the results obtained from different priors are compared. The results show that the Bayesian method provides estimations that are more consistent with the test data and has better robustness in given sets of different sample selections, especially in small sample size.