个旧锡矿区地质数据的极值分布和应用研究
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摘要
本文以地质异常理论为基础的“三联式”成矿预测理论为指导,以个旧锡矿区为示范研究区,针对矿产资源勘查面临的“三难”:难识别、难发现、难开发等问题,在系统研究区域成矿地质背景及深入研究、揭示和分析地质数据的基础上,把数学和地质相结合,把极值理论(Extreme Value Theoy,EVT)应用于个旧锡矿区地质数据的极值统计分布和应用研究,用极值分析的理论和方法进行了异常数据的识别、提取和圈定,对广义帕累托分布(Generalized Pareto Distribution,GPD)模型中的参数估计方法进行了研究和比较,对超过阈值的异常数据建立了GPD模型等方面做了理论和应用方面的研究。
本文重点介绍了极值理论中阈值法(Peak Over Threshold,POT)确定异常下限的理论和方法。区别于经验法,提出了一个定量确定阈值的方法,并把POT法确定异常下限与统计学方法X + 2S进行了比较研究。对GPD模型的参数确定方法进行了研究。比较了几种主要参数估计方法在确定GPD模型中参数时的特点及优缺点。针对确定GPD模型中参数极大似然估计法所得到似然方程是非线性的超越方程没有解析解的特点,提出了一个具有解析解的近似极大似然估计法,详细地推导了近似的计算公式。并强调了在使用时需注意的问题。把极值理论用于个旧东区,对2003年Pb、Zn、Ag、Sn、Mn五种元素的化探数据进行了实证研究,得到的主要结论如下:
1.区别于以往的识别、提取和圈定地质异常的方法。极值理论不仅可以拟合超越异常下限数据的分布,刻画异常发生的概率。而且可以客观的、定量的圈定地质异常,以及描述异常发生的概率,从而刻画了矿产预测的不确定性,而不确定性的定量化和数量化正是矿产定量预测的主要任务之一。
2.通过计算表明:用极值理论的POT法确定的阈值要比用统计学方法X + 2S确定的阈值要小,这样对异常下限的确定上就可以把可能是异常的数据不至于剔除掉,符合成矿预测应最少漏失矿体而又最大减少需进行详细工作面积的原则。
3.计算结果表明Pb元素GPD模型中的参数ξ,σ随阈值的变化情况是:随着阈值的增加,σ随之增大,而ξ随之减少。
Based on the geological anomaly theory and the‘three-components’mineral prediction theory, this paper does research work with the data gathered in the Gejiu Tin ore deposits. Focusing on the problems in the mineral resources exploration such as three difficulties: difficult to identify, difficult to find, difficult to exploit, we combine mathematics with geology to develop our research work based on the systemic study of the background of region mineral forming and on the enough data analysises. We have applied extreme value theory (EVT) to analysis the data of Gejiu Tin ore deposits. We use extreme value theory to recognize, extract and delineate anomaly. The investigations and comparisons are completed for parameter estimation in the model with generalized Pareto distribution (GPD), moreover, the theoretical and applied aspects are completed for building GPD model with anomaly data exceeding the threshold.
The paper mainly describes a method of finding lower limit by using peak over threshold (POT) in extreme value theory, which is different from the experiential method. We provide a method of finding threshold and compare this method with the statistical method X + 2S. Also, we study the parameter estimation in GDP. The characterizations, advantages and disadvantages of several main methods of parameter estimation are given. As we know, the likelihood equation obtained by parameters maximum likelihood estimation in GDP has no analytic solution, so we provide an approximate maximum likelihood estimation and derive its formulae in detail. The formulae can give its analytic solutions and one should pay attention to some problems when applying the formula which has been emphasized in this paper. The following results are mainly achieved after applying extreme value theory to study the data of Pb、Zn、Ag、Sn、Mn in Gejiu Tin ore deposits in 2003.
1. The method in this paper is different from previous the method which is used to recognize, extract and delineate geological anomaly. Extreme value theory not only can fit the distribution which the data exceed the lower limit value of anomaly and characterize the probability of anomaly occurring, but also can delineate geological anomaly objectively and quantitatively,as a result, the indeterminacy in quantitative prediction of minerals is characterized,which is one of assignment in quantitative prediction of minerals.
2. Through the computations, it is shown that the threshold calculated by extreme value theory's POT method is smaller than that calculated by the statistical method X + 2S. Therefore, when determining the lower limit of anomaly, it is possible not to eliminate such data that is probably anomalous. This corresponds with the principle that the mine forming prediction should leak ore-body at least and reduce the required area for detail investigation at most.
3. The results also show that the parameterσwill increase and the parameterξwill decrease when the threshold increases, whereξandσare the parameters in generalized Pareto distribution of Pb.
本文重点介绍了极值理论中阈值法(Peak Over Threshold,POT)确定异常下限的理论和方法。区别于经验法,提出了一个定量确定阈值的方法,并把POT法确定异常下限与统计学方法X + 2S进行了比较研究。对GPD模型的参数确定方法进行了研究。比较了几种主要参数估计方法在确定GPD模型中参数时的特点及优缺点。针对确定GPD模型中参数极大似然估计法所得到似然方程是非线性的超越方程没有解析解的特点,提出了一个具有解析解的近似极大似然估计法,详细地推导了近似的计算公式。并强调了在使用时需注意的问题。把极值理论用于个旧东区,对2003年Pb、Zn、Ag、Sn、Mn五种元素的化探数据进行了实证研究,得到的主要结论如下:
1.区别于以往的识别、提取和圈定地质异常的方法。极值理论不仅可以拟合超越异常下限数据的分布,刻画异常发生的概率。而且可以客观的、定量的圈定地质异常,以及描述异常发生的概率,从而刻画了矿产预测的不确定性,而不确定性的定量化和数量化正是矿产定量预测的主要任务之一。
2.通过计算表明:用极值理论的POT法确定的阈值要比用统计学方法X + 2S确定的阈值要小,这样对异常下限的确定上就可以把可能是异常的数据不至于剔除掉,符合成矿预测应最少漏失矿体而又最大减少需进行详细工作面积的原则。
3.计算结果表明Pb元素GPD模型中的参数ξ,σ随阈值的变化情况是:随着阈值的增加,σ随之增大,而ξ随之减少。
Based on the geological anomaly theory and the‘three-components’mineral prediction theory, this paper does research work with the data gathered in the Gejiu Tin ore deposits. Focusing on the problems in the mineral resources exploration such as three difficulties: difficult to identify, difficult to find, difficult to exploit, we combine mathematics with geology to develop our research work based on the systemic study of the background of region mineral forming and on the enough data analysises. We have applied extreme value theory (EVT) to analysis the data of Gejiu Tin ore deposits. We use extreme value theory to recognize, extract and delineate anomaly. The investigations and comparisons are completed for parameter estimation in the model with generalized Pareto distribution (GPD), moreover, the theoretical and applied aspects are completed for building GPD model with anomaly data exceeding the threshold.
The paper mainly describes a method of finding lower limit by using peak over threshold (POT) in extreme value theory, which is different from the experiential method. We provide a method of finding threshold and compare this method with the statistical method X + 2S. Also, we study the parameter estimation in GDP. The characterizations, advantages and disadvantages of several main methods of parameter estimation are given. As we know, the likelihood equation obtained by parameters maximum likelihood estimation in GDP has no analytic solution, so we provide an approximate maximum likelihood estimation and derive its formulae in detail. The formulae can give its analytic solutions and one should pay attention to some problems when applying the formula which has been emphasized in this paper. The following results are mainly achieved after applying extreme value theory to study the data of Pb、Zn、Ag、Sn、Mn in Gejiu Tin ore deposits in 2003.
1. The method in this paper is different from previous the method which is used to recognize, extract and delineate geological anomaly. Extreme value theory not only can fit the distribution which the data exceed the lower limit value of anomaly and characterize the probability of anomaly occurring, but also can delineate geological anomaly objectively and quantitatively,as a result, the indeterminacy in quantitative prediction of minerals is characterized,which is one of assignment in quantitative prediction of minerals.
2. Through the computations, it is shown that the threshold calculated by extreme value theory's POT method is smaller than that calculated by the statistical method X + 2S. Therefore, when determining the lower limit of anomaly, it is possible not to eliminate such data that is probably anomalous. This corresponds with the principle that the mine forming prediction should leak ore-body at least and reduce the required area for detail investigation at most.
3. The results also show that the parameterσwill increase and the parameterξwill decrease when the threshold increases, whereξandσare the parameters in generalized Pareto distribution of Pb.
引文
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