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1.
An objective replacement method for censored geochemical data   总被引：1，自引：0，他引：1  
Geochemical data are commonly censored, that is, concentrations for some samples are reported as less than or greater than some value. Censored data hampers statistical analysis because certain computational techniques used in statistical analysis require a complete set of uncensored data. We show that the simple substitution method for creating an uncensored dataset, e.g., replacement by3/4 times the detection limit, has serious flaws, and we present an objective method to determine the replacement value. Our basic premise is that the replacement value should equal the mean of the actual values represented by the qualified data. We adapt the maximum likelihood approach (Cohen, 1961) to estimate this mean. This method reproduces the mean and skewness as well or better than a simple substitution method using3/4 of the lower detection limit or3/4 of the upper detection limit. For a small proportion of less than substitutions, a simple-substitution replacement factor of 0.55 is preferable to3/4; for a small proportion of greater than substitutions, a simple-substitution replacement factor of 1.7 is preferable to4/3, provided the resulting replacement value does not exceed 100%. For more than 10% replacement, a mean empirical factor may be used. However, empirically determined simple-substitution replacement factors usually vary among different data sets and are less reliable with more replacements. Therefore, a maximum likelihood method is superior in general. Theoretical and empirical analyses show that true replacement factors for less thans decrease in magnitude with more replacements and larger standard deviation; those for greater thans increase in magnitude with more replacements and larger standard deviation. In contrast to any simple substitution method, the maximum likelihood method reproduces these variations. Using the maximum likelihood method for replacing less thans in our sample data set, correlation coefficients were reasonably accurately estimated in 90% of the cases for as much as 40% replacement and in 60% of the cases for 80% replacement. These results suggest that censored data can be utilized more than is commonly realized.  相似文献   

2.
Earthquake hazard parameters maximum regional magnitude M max and annual activity rate , and the b parameter of the Gutenberg-Richter relation have been evaluated for parts of East Africa. The applied maximum likelihood method permits the combination of both historical and instrumental data available (the catalog used here covers the interval 1880–1979). In addition, the uncertainty involved in magnitude determination and thresholds of completeness were taken into account. The hazard-parameter determination was performed for two study areas corresponding to segments of the eastern and western branches of the East African rift system. The results for these areas show differences that can be partly explained through the characteristics of the data.  相似文献   

3.
The Second-Order Stationary Universal Kriging Model Revisited   总被引：3，自引：0，他引：3  
Universal kriging originally was developed for problems of spatial interpolation if a drift seemed to be justified to model the experimental data. But its use has been questioned in relation to the bias of the estimated underlying variogram (variogram of the residuals), and furthermore universal kriging came to be considered an old-fashioned method after the theory of intrinsic random functions was developed. In this paper the model is reexamined together with methods for handling problems in the inference of parameters. The efficiency of the inference of covariance parameters is shown in terms of bias, variance, and mean square error of the sampling distribution obtained by Monte Carlo simulation for three different estimators (maximum likelihood, bias corrected maximum likelihood, and restricted maximum likelihood). It is shown that unbiased estimates for the covariance parameters may be obtained but if the number of samples is small there can be no guarantee of good estimates (estimates close to the true value) because the sampling variance usually is large. This problem is not specific to the universal kriging model but rather arises in any model where parameters are inferred from experimental data. The validity of the estimates may be evaluated statistically as a risk function as is shown in this paper.  相似文献   

4.
To simultaneously evaluate the decay constant of 40K () and the age of a standard (t std) using isotopic data from geologic materials, we applied a series of statistical methods. The problem of estimating the most probable intercept of many nonlinear curves in and t std space is formulated by an errors-in-variables nonlinear regression model. Then a maximum likelihood method is applied to the model for a point estimate, which is equivalent to the nonlinear least square method when measurement error distributions are Gaussian. Uncertainties and confidence regions of the estimates can be approximated using three methods: the asymptotic normal approximation, the parametric bootstrap method and Bonferroni confidence regions. Five pairs of published data for samples with ages from 2 ka to 4.5 Ga were used to estimate and the age of Fish Canyon sanidine (t FCs). The statistical procedure yields most probable estimates of (5.4755 ± 0.0170 × 10–10 (1)/year) and t FCs (28.269 ± 0.0661 (1) Ma) which are in between previously published values. These results indicate the power of our approach to provide improved constraints on these parameters, although the preliminary nature of some of the input data require further review before the values can be adopted.  相似文献   

5.
Joint estimation of transmissivity (T) and storativity (S) in a confined aquifer is done via maximum likelihood (ML). The differential equation of groundwater flow is discretized by the finite-element method, leading to equation t+x t=u t. Elements of matrices and , as well as estimated covariance matrix of noise termu t, are functions of T and S. By minimizing the negative loglikelihood function corresponding to discretized groundwater flow equation with respect to T and S, ML estimators are obtained. The ML approach is found to yield accurate estimates of T and S (within 9 and 10% of their actual values, respectively) and showed quadratic convergence in Newton's search technique. Prediction of aquifer response, using ML estimators, results in estimated piezometric heads accurate to ±0.5 m from their actual, exact values. Statistical properties of ML estimators are derived and some basic results for statistical inference are given.  相似文献   

6.
Seismic hazard of Egypt   总被引：1，自引：0，他引：1  
Earthquake hazard parameters such as maximum expected magnitude,M max, annual activity rate,, andb value of the Gutenberg-Richter relation have been evaluated for two regions of Egypt. The applied maximum likelihood method permits the combination of both historical and instrumental data. The catalogue used covers earthquakes with magnitude 3 from the time interval 320–1987. The uncertainties in magnitude estimates and threshold of completeness were taken into account. The hazard parameter determination is performed for two study areas. The first area, Gulf of Suez, has higher seismicity level than the second, all other active zones in Egypt.b-values of 1.2 ± 0.1 and 1.0 ± 0.1 are obtained for the two areas, respectively. The number of annually expected earthquakes with magnitude 3 is much larger in the Gulf of Suez, 39 ± 2 than in the other areas, 6.1 ± 0.5. The maximum expected magnitude is calculated to be 6.5 ± 0.4 for a time span of 209 years for the Gulf of Suez and 6.1 ± 0.3 for a time span of 1667 years for the remaining active areas in Egypt. Respective periods of 10 and 20 years were reported for earthquakes of magnitude 5.0 for the two subareas.  相似文献   

7.
This paper introduces the concept of quasi-independence into embedded Markov chains and identifies the maximum likelihood frequency estimates to be used with 2 tests of the null hypothesis of quasi-independence within an embedded Markov chain.  相似文献   

8.
Öncel  A. O.  Alptekin  Ö. 《Natural Hazards》1999,19(1):1-11
In order to investigate the effect of aftershocks on earthquake hazard estimation, earthquake hazard parameters (m, b and Mmax) have been estimated by the maximum likelihood method from the main shocks catalogue and the raw earthquakes catalogue for the North Anatolian Fault Zone (NAFZ). The main shocks catalogue has been compiled from the raw earthquake catalogue by eliminating the aftershocks using the window method. The raw earthquake catalogue consisted of instrumentally detected earthquakes between 1900 and 1992, and historical earthquakes that occurred between 1000–1900. For the events of the mainshock catalogue the Poisson process is valid and for the raw earthquake catalogue it does not fit. The paper demonstrates differences in the hazard outputs if on one hand the main catalogues and on the other hand the raw catalogue is used. The maximum likelihood method which allows the use of the mixed earthquake catalogue containing incomplete (historical) and complete (instrumental) earthquake data is used to determine the earthquake hazard parameters. The maximum regional magnitude (Mmax, the seismic activity rate (m), the mean return period (R) and the b value of the magnitude-frequency relation have been estimated for the 24°–31° E, 31°–41° E, 41°–45° E sections of the North Anatolian Fault Zone from the raw earthquake catalogue and the main shocks catalogue. Our results indicate that inclusion of aftershocks changes the b value and the seismic activity rate m depending on the proportion of aftershocks in a region while it does not significantly effect the value of the maximum regional magnitude since it is related to the maximum observed magnitude. These changes in the earthquake hazard parameters caused the return periods to be over- and underestimated for smaller and larger events, respectively.  相似文献   

9.
Multivariate statistical analyses have been extensively applied to geochemical measurements to analyze and aid interpretation of the data. Estimation of the covariance matrix of multivariate observations is the first task in multivariate analysis. However, geochemical data for the rare elements, especially Ag, Au, and platinum-group elements, usually contain observations the below detection limits. In particular, Instrumental Neutron Activation Analysis (INAA) for the rare elements produces multilevel and possibly extremely high detection limits depending on the sample weight. Traditionally, in applying multivariate analysis to such incomplete data, the observations below detection limits are first substituted, for example, each observation below the detection limit is replaced by a certain percentage of that limit, and then the standard statistical computer packages or techniques are used to obtain the analysis of the data. If a number of samples with observations below detection limits is small, or the detection limits are relatively near zero, the results may be reasonable and most geological interpretations or conclusions are probably valid. In this paper, a new method is proposed to estimate the covariance matrix from a dataset containing observations below multilevel detection limits by using the marginal maximum likelihood estimation (MMLE) method. For each pair of variables, sayY andZ whose observations containing below detection limits, the proposed method consists of three steps: (i) for each variable separately obtaining the marginal MLE for the means and the variances, , , , and forY andZ: (ii) defining new variables by and and lettingA=C+D andB=C–D, and obtaining MLE for variances, and forA andB; (iii) estimating the correlation coefficient YZ by and the covariance YZ by . The procedure is illustrated by using a precious metal geochemical data set from the Fox River Sill, Manitoba, Canada.  相似文献   

10.
Kern(1982)和 Kern 与 Richter(1981)介绍了他们通过高温高压实验测定的、多种岩石在不同温度和压力条件下模拟地震 P 波和 S 波的波速 Vp 和 Vs,这些资料对于探索地壳内的岩石矿物组成和各种物理性质是十分宝贵的。我们知道,Vp 和 Vs 既依赖于温度,也依赖于压力。  相似文献   

11.
The maximum likelihood estimation of earthquake hazard parameters (maximum regional magnitudem max, activity rate , and theb parameter in the Gutenberg-Richter distribution) is extended to the cases of incomplete and uncertain data. The method accepts mixed data containing only large (extreme) events and a variable quality of complete data with different threshold magnitude values. Uncertainty of earthquake magnitude is specified by two values, the lower and upper magnitude limits. It is assumed that such an interval contains the real unknown magnitude. The proposed approach allows the combination of different quality catalog parts, e.g. those where the assignment of magnitude is questionable and those with magnitudes precisely determined.As an illustration of the method, the seismic hazard analysis for western Norway and adjacent sea area (4–8°E, 58–64°N) is presented on the basis of the strongest earthquakes felt during the period 1831–1889 and three complete catalog parts, covering the period 1890–1987.Paper presented at the 21st General Assembly of the European Seismological Commission held in Sofia, 1988.On leave from Institute of Geophysics, Polish Academy of Sciences, Warsaw, Poland.  相似文献   

12.
Usual methods for fitting a straight line, Y = + X,to data fail if the independent variable Xis subject to error. The problem is further complicated if there is no strong reason for selecting one of the two variables as independent; neither of the two lines may be correct. This review article discusses the maximum likelihood estimators of and under functional and structural models. These models involve differing assumptions about the statistical distributions of the dependent and independent variables. In addition, least-squares procedures are also considered. All these methods lead to the same result, a quadratic equation which can be solved to give an estimate of . This result requires knowledge of the ratio of the error variances, = 2/2, where 2 is the variance of the Yresiduals and 2 is the variance of the X residuals. If 2 and 2 are unknown, estimates of can be difficult to obtain. If replicate sampling was employed, estimates of the variances can be made, and then of .Contribution Number 1 of the series of review articles by the Mathematical Geologists of the United States.  相似文献   

13.
Summary A new concept of feature size range of a roughness profile is introduced in the paper. It is shown that this feature size range plays an important role in estimating the fractal dimension,D, accurately using the divider method. Discussions are given to indicate the difficulty of using both the divider and the box methods in estimatingD accurately for self-affine profiles. The line scaling method's capability in quantifying roughness of natural rock joint profiles, which may be self-affine, is explored. Fractional Brownian profiles (self-affine profiles) with and without global trends were generated using known values ofD, input standard deviation, , and global trend angles. For different values of the input parameter of the line scaling method (step sizea 0),D and another associated fractal parameterC were calculated for the aforementioned profiles. Suitable ranges fora 0 were estimated to obtain computedD within ±10% of theD used for the generation. Minimum and maximum feature sizes of the profiles were defined and calculated. The feature size range was found to increase with increasingD and , in addition to being dependent on the total horizontal length of the profile and the total number of data points in the profile. The suitable range fora 0 was found to depend on bothD and , and then, in turn, on the feature size range, indicating the importance of calculating feature size range for roughness profiles to obtain accurate estimates for the fractal parameters. Procedures are given to estimate the suitablea 0 range for a given natural rock joint profile to use with the line scaling method in estimating fractal parameters within ±10% error. Results indicate the importance of removal of global trends of roughness profiles to obtain accurate estimates for the fractal parameters. The parametersC andD are recommended to use with the line scaling method in quantifying stationary roughness. In addition, one or more parameters should be used to quantify the non-stationary part of roughness, if it exists. The estimatedC was found to depend on bothD and and seems to have potential to capture the scale effect of roughness profiles.  相似文献   

14.
Summary Palaeotemperature determinations were made by the oxygen isotope method on 89 Belemnoids from the Schwäb. Jura, South Germany. The data derived show a well-defined maximum in the Upper Toarcian and Lower Bajocian (up to 29 C). This maximum is followed by a decline in the Upper Bajocian times (13.2 C –18.1 C). In the Lower and Middle Lias, as in the Malm, temperatures between 18 C and 24 C are prevalent.It is indicated that biological-ecological factors may change the O18/O36 ratio of the carbonate of the rostra and consequently the palaeotemperature results.  相似文献   

15.
Field tests of hydraulic conductivity (e.g., injection test, pumping test, etc.) in low permeability formations are subject to censoring due to the detection limit of the instruments used. An iterative method of estimating the mean and variance of hydraulic conductivity data with a presumed log-normal distribution function is presented. This method accounts for the data that are actually below the lower detection limit (called truncated data) and thus gives distribution parameters that are more representative for the underlying distribution. The proposed method is then tested on two simulated normally distributed random datasets having different variances. The results show that the means and variances estimated by the proposed method are very accurate. Finally, the method is used to estimate the mean and variance of hydraulic conductivity data from single hole water injection tests in a fractured geological formation.  相似文献   

16.
Over sixty syntectonic deformation experiments in uniaxial compression have been done on fine-grained limestones in the stability fields of calcite I, calcite II and aragonite. X-ray techniques and spherical harmonic analysis of the data were used to determine preferred orientation quantitatively, and inverse pole-figures were derived for these axially symmetric specimens. They display in most cases strong preferred orientation which varies as a function of the experimental conditions, mainly temperature and pressure. At temperatures below 350° C recrystallization is lacking and flattened grains indicate that translation, twin gliding and kinking have been the dominant deformation mechanisms. The inverse pole-figure shows a maximum at c with a shoulder towards or a second maximum at e. This is in agreement with preferred orientation observed in experimentally deformed Yule marble and can be explained as the product of dominant twin gliding on e and translation gliding on r (Turner et al., 1956). At high temperatures (900–1000° C) strong grain growth (from 4 to 50 microns) indicates that the fabric recrystallized. Grains are equidimensional and clear with a marble-like texture. The inverse pole-figure shows a single maximum at r, and c-axes are oriented in a small circle around the axis of compression, 1. Such a pattern of preferred orientation would be expected on thermodynamic grounds assuming that recrystallized grains will be oriented in such a way that the strain energy is a maximum (e.g. MacDonald, 1960). Decrease in confining pressure caused a decrease of the maximum at c and the formation of a secondary maximum at highangle positive rhombs in the inverse pole-figure. This can be interpreted as r translation dominating over e twinning. In all deformation experiments an equilibrium in preferred orientation was reached after 20 percent shortening. The strength of preferred orientation decreased with increasing temperature. Aragonite was produced within its hydrostatic stability field at temperatures above 500° C. Close to the phase boundary, coarse-grained textures showed preferred orientation with poles to (010) parallel to 1. At higher pressures the fabric is fine-grained and [001] is aligned parallel to 1. Evidence is given that the phase change from calcite to aragonite in these deformation experiments is a diffusive and not a martensitic transformation.Publication No. 1043, Institute of Geophysics and Planetary Physics, University of California, Los Angeles, California.  相似文献   

17.
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18.
We report the results of a Sm-Nd isotopic study of whole-rock samples from the Chamrousse ophiolite complex, in the Variscan basement of the Western Alps, France. Our data point to a 497±24 (2) Ma age, in excellent agreement with U-Pb results (Ménot et al. 1984) on subordinate plagiogranites. Strong isotopic heterogeneity and time evolution of magmas from relatively less depleted ( Nd= +5) to very depleted ( Nd= +9) sources are also implied. Available data favour a mixing model between N-MORB and E-MORB type end-members, but a third component with distinct supra-subduction zone affinity is also required. These results preclude any tectonic setting such as intracontinental rift, ensialic marginal basin, or major ocean ridge, but substantiate an oceanic back-arc environment. They also demonstrate that the well-documented Upper Cambrian-Lower Ordovician extensional phase here reached a true oceanic stage.  相似文献   

19.
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20.
To investigate the equation of state of -MnS at high pressure and the possibility of a phase transition, the compression curve was measured at 298 K from 0 to 21 GPa using powder x-ray diffraction with a diamond anvil cell. The compression data are fit to a thirdorder Birch-Murnaghan equation of state, with parameters K 0 = 72(2) GPa and K 0 = 4.2(13). To compare present results with previous work, the data sets from three previous investigations (Clendenen and Drickamer 1966; Wakabayashi et al. 1968; Kraft and Greuling 1988) are refit to a Birch-Murnaghan equation of state. In the low pressure region (P < 10=" gpa),=" the=" results=" of=" clendenen=" and=" drickamer=" (1966)=" agree=" with=" the=" present=" data;=" however=" the=" results=" of=" wakbayashi=" et=" al.=" (1968)=" differ=" by=" more=" than=" 10%.=" a=" greater=" discrepancy=" between=" the=" present=" and=" previous=" results=" occurs=" above=" 10=" gpa.=" kraft=" and=" greuling=" (1988)=" reported=" a=" structure=" transition=" at=" 7=" gpa,=" and=" clendenen=" and=" drickamer=" (1966)=" observed=" a=" structure=" distortion=" at=" approximately=" 10=" gpa;=" the=" present=" data=" show=" no=" evidence=" of=" either=" transition,=" and=" are=" well=" fit=" by=" a=" single=" equation=" of=" state=" from=" 0=" to=" 21=" gpa.=" nonhydrostatic=" stress=" is=" discussed=" as=" one=" possibility=" for=" the=">  相似文献   

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