基于概率密度演化理论的重庆市降雨概率结构研究
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摘要
降雨作为市政工程和土木工程领域重要的灾害荷载之一,其概率结构对相应随机系统的分析有着至关重要的作用。然而,常用的概率模型往往不能很好地描述降雨的概率结构。为此,针对线性虚拟随机过程的广义密度演化方程及其形式解析解,导出了可直接用于随机静力系统分析的概率密度变换解,并发展了δ序列逼近算法。若将随机数据视为自映射系统,则上述方法可方便用于随机数据的概率结构分析。采用上述方法,该文分析了重庆市降雨的概率结构,获得了最大日降雨量概率密度函数的近似解,并由等间距的频数直方图和等频数直方图以及经验累积分布函数验证了近似解的准确性、可信性。遗憾的是,真实的概率密度函数复杂而不便于实用。因此,该文建议了一种描述复杂概率结构的线性组合模型,并通过试算法给出了最大日降雨量的实用概率模型,为后续的随机系统分析奠定基础。
As one of hazard actions in the field of municipal engineering and civil engineering,the probabilistic structure of rainfall is very important for the analysis of stochastic systems involved in rainfall.However,it is difficult for modeling the probabilistic structure of rainfall by a traditional probability density function,sometimes it is impossible.In this work,a new technique coming from probability density evolution theory is proposed to describe the probabilistic structure of rainfall.Based on the analytical solution for generalized density evolution equations of a virtual random process,the transiting solution of a probability density function and its approximation via a family of δ sequences are derived.This transiting solution promotes probability density evolution theory from dynamical systems to static systems.Obviously,it is applicable to the probability analysis of random data,which can be viewed as a self-mapping static system.Therefore,the probabilistic structure of rainfall in Chongqing can be obtained by this method,verified by the frequency histogram of an equal interval,a frequency histogram of equal probability and empirical cumulative distribution functions.In order to build up a simple and practical model for the complicated,a class of linear combined model based on traditional probability density functions is put forward,and linear combined models for maximum daily rainfall are obtained by try and error.
As one of hazard actions in the field of municipal engineering and civil engineering,the probabilistic structure of rainfall is very important for the analysis of stochastic systems involved in rainfall.However,it is difficult for modeling the probabilistic structure of rainfall by a traditional probability density function,sometimes it is impossible.In this work,a new technique coming from probability density evolution theory is proposed to describe the probabilistic structure of rainfall.Based on the analytical solution for generalized density evolution equations of a virtual random process,the transiting solution of a probability density function and its approximation via a family of δ sequences are derived.This transiting solution promotes probability density evolution theory from dynamical systems to static systems.Obviously,it is applicable to the probability analysis of random data,which can be viewed as a self-mapping static system.Therefore,the probabilistic structure of rainfall in Chongqing can be obtained by this method,verified by the frequency histogram of an equal interval,a frequency histogram of equal probability and empirical cumulative distribution functions.In order to build up a simple and practical model for the complicated,a class of linear combined model based on traditional probability density functions is put forward,and linear combined models for maximum daily rainfall are obtained by try and error.
引文
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[17]向斌,陈文,胡婷婷.酸雨地区混凝土腐蚀与防护[J].全面腐蚀控制,2009,23(6):41―46.Xiang Bin,Chen Wen,Hu Tingting.Corrosion andprotection of concrete in acid rain area[J].TotalCorrosion Control,2009,23(6):41―46.(in Chinese)
[18]Xie S,Qi L,Zhou D.Investigation of the effects of acidrain on the deterioration of cement concrete usingaccelerated tests established in laboratory[J].Atmospheric Environment,2004,38:4457―4466.
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[20]巴金,汤洁,王淑凤,等.重庆地区近10年酸雨时空分布和季节变化特征分析[J].气象,2008,34(9):81―88.Ba Jin,Tang Jie,Wang Shufeng,et al.Analysis ofspatio-temporal distribution and seasonality of acid rainin Chongqing from 1997 to 2006[J].MeteorologicalMonthly,2008,34(9):81―88.(in Chinese)
[21]畅航.重庆地区输电铁塔腐蚀与防腐蚀研究[D].重庆:重庆大学,2002.Chang Hang.Research on corrosion and anticorrosion ofiron tower for transporting electricity in Chongqing area[D].Chongqing:Chongqing University,2002.(inChinese)
[22]Bendat J S,Piersol A G.Random data:Analysis andmeasurement procedures[M].3rd ed.New York:JohnWiley&Sons Inc.,2000:99―108.
[23]李杰,陈建兵.随机动力系统中的广义密度演化方程[J].自然科学进展,2006,16(6):712―719.Li Jie,Chen Jianbing.The probability density evolutionanalysis of stochastic dynamic system[J].Progress inNatural Science,2006,16(6):712―719.(in Chinese)
[24]Li J,Chen J B.The principle of preservation ofprobability and the generalized density evolutionequation[J].Structural Safety,2008,30:65―77.
[25]Chen J B,Li J.The extreme value distribution andreliability of nonlinear stochastic structures[J].Earthquake Engineering&Engineering Vibration,2005,4(2):275―286.
[26]李杰,陈建兵,张琳琳,等.中国大陆地区年最大平均风速的概率密度函数[J].自然灾害学报,2006,15(5):76―82.Li Jie,Chen Jianbing,Zhang Linlin,et al.Probabilitydensity function of yearly maximum average wind speedin mainland of China[J].Journal of Natural Disasters,2006,15(5):76―82.(in Chinese)
[27]Farlow S J.Partial differential equations for scientistsand engineers[M].New York:Dover Publications Inc.,1993:205―212.
[28]范文亮,李杰.广义密度演化方程的δ函数序列解法[J].力学学报,2009,41(3):398―409.Fan Wenliang,Li Jie.Solution of generalized densityevolution equation via a family ofδsequences[J].Chinese Journal of Theoretical and Applied Mechanics,2009,41(3):398―409.(in Chinese)
[29]盖尔芳特,希洛夫.广义函数Ⅰ——广义函数及其运算[M].林坚冰,译.北京:科学出版社,1965:33―38.Gelfand I M,Shilov G E.Generalized function,VolumeⅠ——generalized function and its operation[M].Translated by Lin Jianbin.Beijing:Science Press,1965:33―38.(in Chinese)
[30]Gentle J E.Elements of computational statistics[M].New York:Springer-Verlag New York,Inc.,2002:204―212.
[2]胡聿贤.地震工程学[M].第2版.北京:地震出版社,2006:1―566.Hu Yuxian.Earthquake engineering[M].2nd ed.Beijing:Earthquake Press,2006:1―566.(in Chinese)
[3]Simiu E,Scanlan R H.Wind effects on structures[M].2nd ed.New York:John Wiley and Sons Inc.,1986:1―589.
[4]欧进萍,王光远.结构随机振动[M].北京:高等教育出版社,1998:99―380.Ou Jinping,Wang Guangyuan.Structural randomvibration[M].Beijing:Higher Education Press,1998:99―380.(in Chinese)
[5]Soong T T,Grigoriu M.Random vibration of mechanicaland structural systems[M].New Jersey:Prentice Hall,1993:64―337.
[6]解以扬,李大鸣,李培彦,等.城市暴雨内涝数学模型的研究与应用[J].水科学进展,2005,16(3):384―390.Xie Yiyang,Li Daming,Li Peiyan,et al.Research andapplication of the mathematic model for urban rainstormwater logging[J].Advances in Water Science,2005,16(3):384―390.(in Chinese)
[7]赵彩萍,荆肖军,李艳红,等.城市暴雨内涝预报研究[J].科技情报开发与经济,2008,18(29):114―116,121.Zhao Caiping,Jin Xiaojun,Li Yanhong,et al.Researchon the forecasting of water logging caused by urbanrainstorm[J].Sci-Tech Information Development&Economy,2008,18(29):114―116,121.(in Chinese)
[8]陈能志,林阗,汪裕丰.福州市中心城区内涝治理研究[J].中国水利,2008(9):40―42.Chen Nengzhi,Lin Tian,Wang Yufeng.Study on waterlogging control in downtown areas of Fuzhou city[J].China Water Resources,2008(9):40―42.(in Chinese)
[9]张友谊,胡卸文,朱海勇.滑坡与降雨关系研究展望[J].自然灾害学报,2007,16(1):104―108.Zhang Youyi,Hu Xiewen,Zhu Haiyong.Prospect ofresearch on relationship between landslide and rainfall[J].Journal of Natural Disasters,2007,16(1):104―108.(in Chinese)
[10]张珍,李世海,马力.重庆地区滑坡与降雨关系的概率分析[J].岩石力学与工程学报,2005,24(17):3185―3191.Zhang Zhen,Li Shihai,Ma Li.Probability analysis ofrelationship between landslide and rainfall in Chongqingarea[J].Chinese Journal of Rock Mechanics andEngineering,2005,24(17):3185―3191.(in Chinese)
[11]Polemio M,Sdao F.The role of rainfall in the landslidehazard:The case of the Avigliano urban area[J].Engineering Geology,1999,53:297―309.
[12]Dai F C,Lee C F.Frequency-volume relation andprediction of rainfall-induced landslides[J].EngineeringGeology,2001,59:253―266.
[13]Chen H,Lee C F.A dynamic model for rainfall-inducedlandslides on natural slopes[J].Geomorphology,2003,51:269―288.
[14]张学元,韩恩厚,李洪锡.中国的酸雨对材料腐蚀的经济损失估算[J].中国腐蚀与防护学报,2002,22(5):316―319.Zhang Xuanyuan,Han Enhou,Li Hongxi.Estimation ofthe corrosion losses by the acidic rain in China[J].Journal of Chinese Society for Corrosion and Protection,2002,22(5):316―319.(in Chinese)
[15]安百刚,张学元,韩恩厚,等.A3钢在模拟降雨环境下的腐蚀和冲刷行为研究[J].金属学报,2002,38(7):755―759.An Baigang,Zhang Xueyuan,Han Enhou,et al.Thebehavior of corrosion and runoff of A3 steel in artificialrainwater[J].Acta Metallurgica Sinica,2002,38(7):755―759.(in Chinese)
[16]黎学明,刘强,孔令峰,等.模拟酸雨溶液中应力对镀锌钢绞线腐蚀行为影响[J].腐蚀科学与防腐技术,2008,20(1):44―46.Li Xueming,Liu Qiang,Kong Lingfeng,et al.Corrosionbehavior of galvanized steel under stress in simulationacid rain solution[J].Corrosion Science and ProtectionTechnology,2008,20(1):44―46.(in Chinese)
[17]向斌,陈文,胡婷婷.酸雨地区混凝土腐蚀与防护[J].全面腐蚀控制,2009,23(6):41―46.Xiang Bin,Chen Wen,Hu Tingting.Corrosion andprotection of concrete in acid rain area[J].TotalCorrosion Control,2009,23(6):41―46.(in Chinese)
[18]Xie S,Qi L,Zhou D.Investigation of the effects of acidrain on the deterioration of cement concrete usingaccelerated tests established in laboratory[J].Atmospheric Environment,2004,38:4457―4466.
[19]Mori K,Tachi K,Muramatsu M,et al.Mechanism ofacid rain etching of acrylic–melamine coatings[J].Progress in Organic Coatings,1999,36:34―38.
[20]巴金,汤洁,王淑凤,等.重庆地区近10年酸雨时空分布和季节变化特征分析[J].气象,2008,34(9):81―88.Ba Jin,Tang Jie,Wang Shufeng,et al.Analysis ofspatio-temporal distribution and seasonality of acid rainin Chongqing from 1997 to 2006[J].MeteorologicalMonthly,2008,34(9):81―88.(in Chinese)
[21]畅航.重庆地区输电铁塔腐蚀与防腐蚀研究[D].重庆:重庆大学,2002.Chang Hang.Research on corrosion and anticorrosion ofiron tower for transporting electricity in Chongqing area[D].Chongqing:Chongqing University,2002.(inChinese)
[22]Bendat J S,Piersol A G.Random data:Analysis andmeasurement procedures[M].3rd ed.New York:JohnWiley&Sons Inc.,2000:99―108.
[23]李杰,陈建兵.随机动力系统中的广义密度演化方程[J].自然科学进展,2006,16(6):712―719.Li Jie,Chen Jianbing.The probability density evolutionanalysis of stochastic dynamic system[J].Progress inNatural Science,2006,16(6):712―719.(in Chinese)
[24]Li J,Chen J B.The principle of preservation ofprobability and the generalized density evolutionequation[J].Structural Safety,2008,30:65―77.
[25]Chen J B,Li J.The extreme value distribution andreliability of nonlinear stochastic structures[J].Earthquake Engineering&Engineering Vibration,2005,4(2):275―286.
[26]李杰,陈建兵,张琳琳,等.中国大陆地区年最大平均风速的概率密度函数[J].自然灾害学报,2006,15(5):76―82.Li Jie,Chen Jianbing,Zhang Linlin,et al.Probabilitydensity function of yearly maximum average wind speedin mainland of China[J].Journal of Natural Disasters,2006,15(5):76―82.(in Chinese)
[27]Farlow S J.Partial differential equations for scientistsand engineers[M].New York:Dover Publications Inc.,1993:205―212.
[28]范文亮,李杰.广义密度演化方程的δ函数序列解法[J].力学学报,2009,41(3):398―409.Fan Wenliang,Li Jie.Solution of generalized densityevolution equation via a family ofδsequences[J].Chinese Journal of Theoretical and Applied Mechanics,2009,41(3):398―409.(in Chinese)
[29]盖尔芳特,希洛夫.广义函数Ⅰ——广义函数及其运算[M].林坚冰,译.北京:科学出版社,1965:33―38.Gelfand I M,Shilov G E.Generalized function,VolumeⅠ——generalized function and its operation[M].Translated by Lin Jianbin.Beijing:Science Press,1965:33―38.(in Chinese)
[30]Gentle J E.Elements of computational statistics[M].New York:Springer-Verlag New York,Inc.,2002:204―212.
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