河相关系的随机微分方程建模与研究
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摘要
由气候变化等环境突变引起的水沙输入条件和边界条件的不确定性给河流地形特征的考察与模拟增加了困难。本文通过建立随机微分方程,研究了典型的河相关系特征变量(包括比降、河宽、水深、流速)随时间变化的概率分布演化规律。随机方程的随机输入项分别由3种噪声模型进行模拟,包括单独的高斯白噪声、组合的高斯白噪声加泊松噪声,和组合的分数白噪声加泊松噪声模型。方程中的未知参数使用一种复合型的极大似然非参数估计法进行估计。使用蒙特卡洛方法将方程应用于黄河下游高村-孙口段,结果较好地展示了河相关系对随机扰动的潜在响应,特别是,计算的随机平均值与测量值有较好的同步度。通过模型比较发现,既能反映非线性又能反映突变性特征的分数-泊松扩散模型是较适合模拟河相关系随机演化的模型,以此为基础的河流功率可作为系统性评估河流动态演化特征的优选指标。本文提出的分析河相关系的新方法,可根据指定精度用于设计和监测河流系统,既有理论价值又有实际意义。
Uncertainty in flow-sediment input and channel boundary of restriction caused by environmental change(such as clclimatic events) pose difficulties for the accurate acquisition of information on river morphology dynamics. In this study,a set of stable stochastic differential equations(SDEs) are developed to simulate the dynamic probability distributions of typical hydraulic geometry variables represented by slope,width,depth and velocity with varying bankfull discharge at certain moment in river system. The random parts of each equation are modeled based on single Gaussian white noise and further on compound Gaussian/Fractional white noise plus Poisson noise. Consistent estimate of the SDEs parameters are conducted using a composite nonparametric MLE method. The stochastic models are examined with Monte Carlo simulation in a lower Yellow River case,and results successfully reveal the potential responses of hydraulic geometries to stochastic disturbance,and especially,the average trends mainly run to synchronize with the measured values. Comparisons among the three models confirm the advantage of Fractional jump-diffusion model,and through further discussion,stream power on the basis of such model is concluded as the better systematic measure of river dynamics. The proposed stochastic approach is new to the field of fluvial relationships,and its application could help to design and monitor river system with the specified accuracy requirements.
Uncertainty in flow-sediment input and channel boundary of restriction caused by environmental change(such as clclimatic events) pose difficulties for the accurate acquisition of information on river morphology dynamics. In this study,a set of stable stochastic differential equations(SDEs) are developed to simulate the dynamic probability distributions of typical hydraulic geometry variables represented by slope,width,depth and velocity with varying bankfull discharge at certain moment in river system. The random parts of each equation are modeled based on single Gaussian white noise and further on compound Gaussian/Fractional white noise plus Poisson noise. Consistent estimate of the SDEs parameters are conducted using a composite nonparametric MLE method. The stochastic models are examined with Monte Carlo simulation in a lower Yellow River case,and results successfully reveal the potential responses of hydraulic geometries to stochastic disturbance,and especially,the average trends mainly run to synchronize with the measured values. Comparisons among the three models confirm the advantage of Fractional jump-diffusion model,and through further discussion,stream power on the basis of such model is concluded as the better systematic measure of river dynamics. The proposed stochastic approach is new to the field of fluvial relationships,and its application could help to design and monitor river system with the specified accuracy requirements.
引文
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[12] TEALDI S,CAMPOREALE C,RIDOLFI L. Modeling the impact of river damming on riparian vegetation[J].Journal of Hydrology,2011,396(3):302-312.
[13] MANFREDA S,FIORENTINO M. A stochastic approach for the description of the water balance dynamics in a river basin[J]. Hydrology&Earth System Sciences,2008,12(5):1189-1200.
[14] TSAI C W,MAN C,JUNGSUN O H. Stochastic particle based models for suspended particle movement in sur?face flows[J]. International Journal of Sediment Research,2014,29(2):195-207.
[15] XIA J,LI X,LI T,et al. Response of reach-scale bankfull channel geometry to the altered flow and sediment re?gime in the lower Yellow River[J]. Geomorphology,2014,213(4):255-265.
[16] STEWARDSON M. Hydraulic geometry of stream reaches[J]. Journal of Hydrology,2005,306(1):97-111.
[17]王随继,魏全伟,谭利华,等.山地河流的河相关系及其变化趋势——以怒江、澜沧江和金沙云南河段为例[J].山地学报,2009,27(1):5-13.
[18]黄才安,周济人,赵晓冬.基本河相关系指数的理论研究[J].泥沙研究,2011(6):55-58.
[19]吴保生.冲积河流平滩流量的滞后响应模型[J].水利学报,2008,39(6):680-687.
[20]王光谦,张红武,夏军强.游荡型河流演变及模拟[M].北京:科学出版社,2005.
[21] LEOPOLD L B,WOLMAN M G. River channel patterns:braided,meandering,and straight[M]. US Govern?ment Printing Office,1957.
[22]刘丙军,陈晓宏,王兆礼.河流系统水质时空格局演化研究[J].水文,2007,27(1):8-13.
[23] BLACK F,SCHOLES M. The pricing of options and corporate liabilities[J]. Journal of Political Economy,1973,81(3):637-654.
[24] COX J C,JR J E I,ROSS S A. A theory of the term structure of interest rates[Z]. 2014.
[25] VASICEK O. An equilibrium characterization of the term structure[J]. Journal of Financial and Quantitative Analysis,1977,5(4):177-188.
[26] HIGHAM D J,KLOEDEN P E. Numerical methods for nonlinear stochastic differential equations with jumps[M]. Springer-Verlag New York,Inc.,2005.
[27] CHALMERS,GRAEME D,HIGHAM,et al. Convergence and stability analysis for implicit simulations of sto?chastic differential equations with random jump magnitudes[J]. Discrete and Continuous Dynamical Systems-Se?ries B(DCDS-B),2008,9(1):47-64.
[28] BRUTI-LIBERATI N,PLATEN E. Approximation of jump diffusions in finance and economics[J]. Research Pa?per,2006,29(3/4):283-312.
[29] TSAI C W,MAN C,JUNGSUN O H. Stochastic particle based models for suspended particle movement in sur?face flows[J]. International Journal of Sediment Research,2014,29(2):195-207.
[30] KROESE D P,BOTEV Z I. Spatial Process Simulation[M]//SCHMIDT V. Stochastic Geometry,Spatial Statistics and Random Fields:Models and Algorithms. Cham;Springer International Publishing. 2015:369-404.
[31] YIN Z M. New methods for simulation of Fractional Brownian Motion[J]. Journal of Computational Physics,1996,127(1):66-72.
[32] SORENSEN M. Likelihood Methods for Diffusions with Jumps[M]. Statistical inference in stochastic processes.Marcel Dekker Incorporated. 1991:67-105.
[33] SILVERMAN B W. Density estimation for statistics and data analysis[J]. Technometrics,1986,29(4):495.
[34]邓安军,郭庆超,陈建国.黄河下游驼峰河段演变的特殊性研究[J].人民黄河,2015,37(12):28-31.
[35]郭庆超,黄烈敏,陈建国,等.黄河下游“驼峰”河段的形成和发展[J].泥沙研究,2012(5):38-42.
[36]赵天义,王保民,王广欣,等.黄河下游二级悬河及其“驼峰”河段的治理措施与途径[C]//世界疏浚大会,2010.
[37]江恩惠,韩其为.黄河非平衡输沙典型事例及其研究概述[J].中国水利水电科学研究院学报,2010,8(3):161-165.
[38]夏军强,吴保生,王艳平.近期黄河下游河床调整过程及特点[J].水科学进展,2008,19(3):301-308.
[39]水利电力部黄河水利委员会.黄河流域下游河道水文观测资料(1951年至1971年:铁道至河道)[Z].1951.
[40] MERTON R C. Option pricing when underlying stock returns are discontinuous[J]. Journal of Financial Econom?ics,1976,3(1):125-144.
[41] MANDELBROT B B,AIZENMAN M. Fractals:form,chance,and dimension[J]. Leonardo,1979,12(5):65-66.
[42]张红武.黄河下游洪水模型相似律的研究[D].北京:清华大学,1995.
[2] ODGAARD A J. River-Meander Model. I:Development[J]. Journal of Hydraulic Engineering,1989,115(11):1433-1450.
[3] IKEDA S,PARKER G. Linear Theory of River Meanders[M]//River Meandering. American Geophysical Union,2013:181-213.
[4] CAMPOREALE C,PERONA P,PORPORATO A,et al. Hierarchy of models for meandering rivers and related morphodynamic processes[J]. Reviews of Geophysics,2007,45(1):446-447.
[5] WOLMAN M G,RAN G. Relative scales of time and effectiveness of climate in watershed geomorphology[J].Earth Surface Processes,1978,3(2):189-208.
[6] GRAF W L. Catastrophe theory as a model for change in fluvial systems[Z]. Adjustments of the Fluvial System,1979:13-32.
[7] ANQUETIN S,BRAUD I,VANNIER O,et al. Sensitivity of the hydrological response to the variability of rain?fall fields and soils for the Gard 2002 flash-flood event[J]. Journal of Hydrology,2010,394(1/2):134-147.
[8] MORENO A,VALERO-GARC S B L,GONZ LEZ-SAMP RIZ P,et al. Flood response to rainfall variability dur?ing the last 2000 years inferred from the Taravilla Lake record(Central Iberian Range,Spain)[J]. Journal of Pa?leolimnology,2008,40(3):943-961.
[9] ZOCCATELLI D,BORGA M,CHIRICO G B,et al. The relative role of hillslope and river network routing in the hydrologic response to spatially variable rainfall fields[J]. Journal of Hydrology,2015,531:349-359.
[10] WASIMI S A,MONDAL M S. Periodic transfer function-noise model for forecasting[J]. Journal of Hydrologic Engineering,2005,10(5):353-362.
[11] BOANO F,REVELLI R,RIDOLFI L. Stochastic modelling of DO and BOD components in a stream with random inputs[J]. Advances in Water Resources,2006,29(9):1341-1350.
[12] TEALDI S,CAMPOREALE C,RIDOLFI L. Modeling the impact of river damming on riparian vegetation[J].Journal of Hydrology,2011,396(3):302-312.
[13] MANFREDA S,FIORENTINO M. A stochastic approach for the description of the water balance dynamics in a river basin[J]. Hydrology&Earth System Sciences,2008,12(5):1189-1200.
[14] TSAI C W,MAN C,JUNGSUN O H. Stochastic particle based models for suspended particle movement in sur?face flows[J]. International Journal of Sediment Research,2014,29(2):195-207.
[15] XIA J,LI X,LI T,et al. Response of reach-scale bankfull channel geometry to the altered flow and sediment re?gime in the lower Yellow River[J]. Geomorphology,2014,213(4):255-265.
[16] STEWARDSON M. Hydraulic geometry of stream reaches[J]. Journal of Hydrology,2005,306(1):97-111.
[17]王随继,魏全伟,谭利华,等.山地河流的河相关系及其变化趋势——以怒江、澜沧江和金沙云南河段为例[J].山地学报,2009,27(1):5-13.
[18]黄才安,周济人,赵晓冬.基本河相关系指数的理论研究[J].泥沙研究,2011(6):55-58.
[19]吴保生.冲积河流平滩流量的滞后响应模型[J].水利学报,2008,39(6):680-687.
[20]王光谦,张红武,夏军强.游荡型河流演变及模拟[M].北京:科学出版社,2005.
[21] LEOPOLD L B,WOLMAN M G. River channel patterns:braided,meandering,and straight[M]. US Govern?ment Printing Office,1957.
[22]刘丙军,陈晓宏,王兆礼.河流系统水质时空格局演化研究[J].水文,2007,27(1):8-13.
[23] BLACK F,SCHOLES M. The pricing of options and corporate liabilities[J]. Journal of Political Economy,1973,81(3):637-654.
[24] COX J C,JR J E I,ROSS S A. A theory of the term structure of interest rates[Z]. 2014.
[25] VASICEK O. An equilibrium characterization of the term structure[J]. Journal of Financial and Quantitative Analysis,1977,5(4):177-188.
[26] HIGHAM D J,KLOEDEN P E. Numerical methods for nonlinear stochastic differential equations with jumps[M]. Springer-Verlag New York,Inc.,2005.
[27] CHALMERS,GRAEME D,HIGHAM,et al. Convergence and stability analysis for implicit simulations of sto?chastic differential equations with random jump magnitudes[J]. Discrete and Continuous Dynamical Systems-Se?ries B(DCDS-B),2008,9(1):47-64.
[28] BRUTI-LIBERATI N,PLATEN E. Approximation of jump diffusions in finance and economics[J]. Research Pa?per,2006,29(3/4):283-312.
[29] TSAI C W,MAN C,JUNGSUN O H. Stochastic particle based models for suspended particle movement in sur?face flows[J]. International Journal of Sediment Research,2014,29(2):195-207.
[30] KROESE D P,BOTEV Z I. Spatial Process Simulation[M]//SCHMIDT V. Stochastic Geometry,Spatial Statistics and Random Fields:Models and Algorithms. Cham;Springer International Publishing. 2015:369-404.
[31] YIN Z M. New methods for simulation of Fractional Brownian Motion[J]. Journal of Computational Physics,1996,127(1):66-72.
[32] SORENSEN M. Likelihood Methods for Diffusions with Jumps[M]. Statistical inference in stochastic processes.Marcel Dekker Incorporated. 1991:67-105.
[33] SILVERMAN B W. Density estimation for statistics and data analysis[J]. Technometrics,1986,29(4):495.
[34]邓安军,郭庆超,陈建国.黄河下游驼峰河段演变的特殊性研究[J].人民黄河,2015,37(12):28-31.
[35]郭庆超,黄烈敏,陈建国,等.黄河下游“驼峰”河段的形成和发展[J].泥沙研究,2012(5):38-42.
[36]赵天义,王保民,王广欣,等.黄河下游二级悬河及其“驼峰”河段的治理措施与途径[C]//世界疏浚大会,2010.
[37]江恩惠,韩其为.黄河非平衡输沙典型事例及其研究概述[J].中国水利水电科学研究院学报,2010,8(3):161-165.
[38]夏军强,吴保生,王艳平.近期黄河下游河床调整过程及特点[J].水科学进展,2008,19(3):301-308.
[39]水利电力部黄河水利委员会.黄河流域下游河道水文观测资料(1951年至1971年:铁道至河道)[Z].1951.
[40] MERTON R C. Option pricing when underlying stock returns are discontinuous[J]. Journal of Financial Econom?ics,1976,3(1):125-144.
[41] MANDELBROT B B,AIZENMAN M. Fractals:form,chance,and dimension[J]. Leonardo,1979,12(5):65-66.
[42]张红武.黄河下游洪水模型相似律的研究[D].北京:清华大学,1995.