Simulation on the stochastic evolution of hydraulic geometry relationships with the stochastic changing bankfull discharges in the Lower Yellow River

doi:10.1007/s11442-020-1758-z

Abstract

Abstract:

Extreme weather is an important noise factor in affecting dynamic access to river morphology information. The response characteristics of river channel on climate disturbances draw us to develop a method to investigate the dynamic evolution of bankfull channel geometries (including the hydraulic geometry variables and bankfull discharges) with stochastic differential equations in this study. Three different forms of random inputs, including single Gaussian white noise and compound Gaussian/Fractional white noise plus Poisson noise, are explored respectively on the basis of the classical deterministic models. The model parameters are consistently estimated by applying a composite nonparametric maximum likelihood estimation (MLE) method. Results of the model application in the Lower Yellow River reveal the potential responses of bankfull channel geometries to climate disturbances in a probabilistic way, and, the calculated 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 based on such a model is concluded as a better systematic measure of river dynamics. The proposed method helps to offer an effective tool for analyzing fluvial relationships and improves the ability of crisis management of river system under varying environment conditions.

Key words: stochastic differential equation, bankfull channel geometry, river system, extreme weather, Lower Yellow River

SONG Xiaolong, ZHONG Deyu, WANG Guangqian. Simulation on the stochastic evolution of hydraulic geometry relationships with the stochastic changing bankfull discharges in the Lower Yellow River[J].Journal of Geographical Sciences, 2020, 30(5): 843-864.

Figures/Tables 15

Figure 1

Table 1

Flood season’s average discharge, suspended sediment concentration, and annual measured bankfull channel geometries along the Gaocun station downwards"

Year	Flood season’s average value (*)			Bankfull discharge Q (m³/s)	Slope S (‰)	Width B (m)	Depth D (m)	Velocity U (m/s)
Year	Discharge Q_f (m³/s)		IS coefficient $\xi_{f}$(kg·s/m⁶)	Bankfull discharge Q (m³/s)	Slope S (‰)	Width B (m)	Depth D (m)	Velocity U (m/s)
1952	2417.073		0.0080	6700	0.120	750	1.48	1.57
1953	2562.967		0.0153	5800	0.112	487.5	2.58	1.25
1954	3531.862		0.0132	5500	0.104	450	3.17	1.13
1955	3218.593		0.0095	5400	0.142	711.5	1.54	1.57
1956	2673.740		0.0154	5420	0.121	652	2.19	0.96
1957	1838.984		0.0197	5300	0.125	620.8	2.20	0.95
1958	4190.626		0.0123	5500	0.124	719.5	1.32	1.07
1959	2070.854		0.0356	6700	0.126	684	1.10	1.47
1960	1092.754		0.0343	6500	0.121	325.5	1.02	0.90
1961	2729.593		0.0061	7200	0.103	445	2.37	1.84
1962	2232.033		0.0079	7800	0.114	581	1.78	1.20
1963	2926.106		0.0065	8500	0.110	584	1.30	1.00
1964	4969.268		0.0052	9500	0.113	1217.5	1.69	1.62
1965	1534.878		0.0123	9800	0.128	614	1.24	1.69
1966	2898.642		0.0176	8500	0.139	967	1.24	1.34
1967	4232.683		0.0091	6000	0.125	957.5	1.80	1.80
1968	3114.821		0.0110	6000	0.126	1618.8	1.16	1.63
1969	1155.35		0.0378	5000	0.117	390	1.87	1.54
1970	1675.976		0.0359	4300	0.124	1067	1.35	1.28
1971	1385.22		0.0336	4300	0.123	922.5	1.21	1.36
1972	1205.667		0.0223	3900	0.121	493.5	0.83	0.92
1973	1938.841		0.0282	3500	0.121	566.6	1.09	0.47
1974	1164.151		0.0271	3370	0.121	610.5	1.22	1.76
1975	3035.675		0.0115	4710	0.118	553.5	1.68	1.31
1976	3137.325		0.0088	6090	0.117	542	1.47	1.59
1977	1627.472		0.0419	6500	0.121	366.1	1.32	1.15
1978	1949.756		0.0262	5500	0.113	404.5	2.23	1.12
1979	1945.122		0.0199	5200	0.108	312.1	3.09	1.38
1980	1183.447		0.0225	4500	0.120	483.5	1.30	1.61
1981	3011.618		0.0114	3900	0.114	412.7	2.25	1.39
1982	2226.911		0.0094	5900	0.123	538	1.53	1.64
1983	3310.407		0.0063	7300	0.117	583.7	2.27	1.58
1984	3127.667	0.0070		7400	0.118	551.5	1.66	1.68
1985	2298.691	0.0110		7600	0.113	527.5	2.92	1.54
1986	1207.065	0.0138		7400	0.115	529.5	1.90	0.57
1987	694.1382	0.0216		6800	0.114	233.1	1.78	0.89
1988	1915.215	0.0244		6400	0.110	353.5	2.58	1.53
1989	1796.545	0.0156		4600	0.111	376.5	2.15	1.49
1990	1233.398	0.0217		4500	0.111	363.2	2.31	1.39
1991	429.735	0.0523		4400	0.121	453.5	1.24	0.90
1992	1168.78	0.0383		3200	0.121	465	1.03	1.07
1993	1285.764	0.0207		3600	0.115	460	1.18	1.25
1994	1221.439	0.0394		3700	0.119	469	1.20	1.22
1995	1013.087	0.0465		3000	0.122	486	0.96	1.09
1996	1357.556	0.0232		2800	0.119	537.5	1.27	1.48
1997	299.5125	0.1589		2750	0.126	410.5	1.10	0.94
1998	899	0.0366		2700	0.122	398	0.95	1.10
1999	779.2927	0.0468		2800	0.121	474	1.07	1.13
2000	443.1463	0.0161		2600	0.121	500.5	1.06	0.91
2001	321.3228	0.0232		2400	0.121	486	1.01	1.26
2002	714.4472	0.0148		2000	0.120	446	1.08	1.20
2003	1300.414	0.0113		2300	0.118	448.5	1.26	0.84
2004	818.2926	0.0239		3600	0.115	439	1.34	0.90
2005	897.536	0.0104		4000	0.115	528	1.38	1.17
2006	806.008	0.0079		4500	0.115	431.5	1.33	1.24
2007	1140.674	0.0059		4700	0.112	491	1.62	1.24
2008	625.040	0.0095		4800	0.113	347.5	1.55	1.40
2009	646.455	0.0041		5000	0.116	515	1.21	1.03
2010	1166.422	0.0045		5300	0.118	518.5	1.05	0.99
2011	934.065	0.0051		5400	0.119	668	1.23	1.04
2012	1438.495	0.0057		5400	0.120	648.5	1.24	1.04
2013	1219.894	0.0076		5800	0.118	533.5	1.27	1.10

Table 1

Table 2

Table 3

Table 4

Table 5

Table 6

Table 7

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Table 8

References 41

[1]	Aït-Sahalia Y, Hansen L P , 2009. Handbook of Financial Econometrics: Tools and Techniques. Elsevier.
[2]	Bauch G D, Hickin E J , 2011. Rate of floodplain reworking in response to increasing storm-induced floods, Squamish River, south-western British Columbia, Canada. Earth Surface Processes & Landforms, 36(7):872-884.
[3]	Black F, Scholes M , 1973. The pricing of options and corporate liabilities. Journal of Political Economy, 81(3):637-654. pmid: 10180222
[4]	Bolla Pittaluga M, Nobile G, Seminara G , 2009. A nonlinear model for river meandering. Water Resources Research, 45:546-550.
[5]	Bruti-Liberati N, Platen E , 2006. Approximation of jump diffusions in finance and economics. Research Paper, 29(3/4):283-312.
[6]	Higham D, Chalmers G , 2017. Convergence and stability analysis for implicit simulations of stochastic differential equations with random jump magnitudes. Discrete and Continuous Dynamical Systems: Series B (DCDS-B), 9(1):47-64.
[7]	Chan K, Karolyi G A, Longstaff F A et al., 1992. An empirical comparison of alternative models of the short-term interest rate. Journal of Finance, 47(3):1209-1227. doi: 10.1111/j.1540-6261.1992.tb04011.x
[8]	Decreusefond L, Üstünel A S , 1999. Stochastic analysis of the Fractional Brownian Motion. Potential Analysis, 10(2):177-214. doi: 10.1023/A:1008634027843
[9]	Deng A J, Guo Q C, Chen J G , 2015. Study on particularity of hump reach evolution in Lower Yellow River. Yellow River, 37(12):28-31. (in Chinese)
[10]	Glasserman P , 2004. Monte Carlo Methods in Financial Engineering. Springer.
[11]	Graf W L , 1979. Catastrophe theory as a model for change in fluvial systems. Adjustments of the Fluvial System, 13-32.
[12]	Guo Q C, Huang L M, Chen J G et al., 2012. Formation and development of hump reach in Lower Yellow River. Journal of Sediment Research, ( 5):38-42. (in Chinese)
[13]	Hansen J, Sato M, Ruedy R et al., 2000. Global warming in the twenty-first century: An alternative scenario. Proceedings of the National Academy of Sciences of the United States of America, 97(18):9875-9880. doi: 10.1073/pnas.170278997 pmid: 10944197
[14]	Harrison L R, Pike A, Boughton D A et al., 2017. Coupled geomorphic and habitat response to a flood pulse revealed by remote sensing. Ecohydrology, 10(5).
[15]	Huggett R , 2007. A history of the systems approach in geomorphology. Géomorphologie Relief Processus Environnement, 13(2):145-158. doi: 10.4000/geomorphologie
[16]	Kroese D P, Botev Z I , 2015. Spatial process simulation. In: Schmidt V (ed.), Stochastic Geometry, Spatial Statistics and Random Fields: Models and Algorithms. Springer International Publishing, Cham, 369-404.
[17]	Leland W E, Taqqu M S, Willinger W et al., 1995. On the self-similar nature of Ethernet traffic (extended version). Acm Sigcomm Computer Communication Review, 25(1):202-213. doi: 10.1145/205447
[18]	Liu J M , 2009. Nonlinear forecasting using nonparametric transfer function models. Wseas Transactions on Business & Economics, 157(6):151-164.
[19]	Luchi R, Pittaluga M B, Seminara G , 2012. Spatial width oscillations in meandering rivers at equilibrium. Water Resources Research, 48(5):375-390.
[20]	Mandelbrot B B, Aizenman M , 1979. Fractals: Form, chance, and dimension. Leonardo, 32(5):65-66.
[21]	Manfreda S, Fiorentino M , 2008. A stochastic approach for the description of the water balance dynamics in a river basin. Hydrology & Earth System Sciences, 12(5):1189-1200.
[22]	Merton R C , 1976. Option pricing when underlying stock returns are discontinuous. Working Papers, 3(1):125-144.
[23]	Muneepeerakul R, Rinaldo A, Rodrigueziturbe I , 2007. Effects of river flow scaling properties on riparian width and vegetation biomass. Water Resources Research, 43(12):W12406.
[24]	Navratil O, Albert M B , 2010. Non-linearity of reach hydraulic geometry relations. Journal of Hydrology, 388(3/4):280-290. doi: 10.1016/j.jhydrol.2010.05.007
[25]	Pittaluga M B, Seminara G , 2011. Nonlinearity and unsteadiness in river meandering: A review of progress in theory and modelling. Earth Surface Processes & Landforms, 36(1):20-38.
[26]	Ramezani C A, Zeng Y , 2007. Maximum likelihood estimation of the double exponential jump-diffusion process. Annals of Finance, 3(4):487-507. doi: 10.1007/s10436-006-0062-y
[27]	Rong S , 2006. Theory of stochastic differential equations with jumps and applications: Mathematical and analytical techniques with applications to engineering. Springer Science &Business Media.
[28]	Sørensen M , 1991. Likelihood methods for diffusions with jumps, Statistical inference in stochastic processes. Marcel Dekker Incorporated, 67-105.
[29]	Silverman B W , 1987. Density estimation for statistics and data analysis. Technometrics, 29(4):495. doi: 10.1002/1097-4679(197310)29:4<495::aid-jclp2270290432>3.0.co;2-7 pmid: 4766726
[30]	Stanton R , 1997. A nonparametric model of term structure dynamics and the market price of interest rate risk. Journal of Finance, 52(5):1973-2002. doi: 10.1111/j.1540-6261.1997.tb02748.x
[31]	Tealdi S, Camporeale C, Ridolfi L , 2011. Modeling the impact of river damming on riparian vegetation. Journal of Hydrology, 396(3):302-312. doi: 10.1016/j.jhydrol.2010.11.016
[32]	Tsai C W, Man C, Jungsun O H , 2014. Stochastic particle based models for suspended particle movement in surface flows. International Journal of Sediment Research, 29(2):195-207.
[33]	Wang Z Y, Lee J H W, Melching C S , 2015. River Dynamics and Integrated River Management. Berlin Heidelberg:Springer.
[34]	Wasimi S A, Mondal M S , 2005. Periodic transfer function-noise model for forecasting. Journal of Hydrologic Engineering, 10(5):353-362. doi: 10.1061/(ASCE)1084-0699(2005)10:5(353)
[35]	Wolman M G, Ran G , 1978. Relative scales of time and effectiveness of climate in watershed geomorphology. Earth Surface Processes and Landforms, 3(2):189-208.
[36]	Wu B, Wang G, Xia J et al., 2008. Response of bankfull discharge to discharge and sediment load in the Lower Yellow River. Geomorphology, 100(3):366-376. doi: 10.1016/j.geomorph.2008.01.007
[37]	Xia J, Li X, Li T et al., 2014. Response of reach-scale bankfull channel geometry to the altered flow and sediment regime in the lower Yellow River. Geomorphology, 213:255-265. doi: 10.1016/j.geomorph.2014.01.017
[38]	Yang C T, Song C C S, Woldenberg M J , 1981. Hydraulic geometry and minimum rate of energy dissipation. Water Resources Research, 17(4):1014-1018. doi: 10.1029/WR017i004p01014
[39]	Yang T, Zhang Q, Chen Y D et al., 2010. A spatial assessment of hydrologic alteration caused by dam construction in the middle and lower Yellow River, China. Hydrological Processes, 22(18):3829-3843. doi: 10.1002/hyp.v22:18
[40]	Yin Z M , 1996. New methods for simulation of fractional brownian motion. Journal of Computational Physics, 127(1):66-72. doi: 10.1006/jcph.1996.0158
[41]	Zhang H , 1995. The study of the law of similarity for models of flood flows of the lower reach of the Yellow River [D]. Beijing: Tsinghua University. (in Chinese)

Estimate	K	b	c	$ \beta$	${{\sigma }_{1}}$	$\gamma $
Mean	76.495	-0.477	0.299	0.213	0.136	0.582
SD	23.833	0.044	0.027	0.018	0.005	0.047

Estimate		Slope (S)	Width (B)	Depth (D)	Velocity (U)
m	Mean	0.184	0.771	0.560	0.424
m	SD	0.038	0.218	0.092	0.068
${{\sigma }_{2}}$	Mean	0.073	0.143	0.276	0.130
${{\sigma }_{2}}$	SD	0.148	0.047	0.069	0.023

Estimate	K	b	c	$\beta $	${{\sigma }_{1}}$	$\gamma $	$\lambda _{u}^{[1]} $	$\lambda _{d}^{[1]} $	$1/\eta _{u}^{\left[ 1 \right]} $	$1/\eta _{d}^{[1]} $
Mean	23.818	-0.505	0.432	0.312	0.118	0.494	0.030	0.020	0.215	0.573
SD	4.795	0.025	0.018	0.002	0.003	0.003	0.001	0.004	0.016	0.047

Estimate		Slope (S)	Width (B)	Depth (D)	Velocity (U)
m	Mean	-0.086	0.264	0.350	0.310
m	SD	0.009	0.043	0.059	0.046
${{\sigma }_{2}}$	Mean	0.075	0.301	0.300	0.160
${{\sigma }_{2}}$	SD	0.014	0.065	0.065	0.055
$\lambda _{u}^{[2]} $	Mean	0.180	0.300	0.311	0.394
$\lambda _{u}^{[2]} $	SD	0.001	0.014	0.029	0.065
$\lambda _{d}^{[2]} $	Mean	0.180	0.300	0.327	0.426
$\lambda _{d}^{[2]} $	SD	0.004	0.026	0.054	0.025
$1/\eta _{u}^{[2]} $	Mean	0.059	0.079	0.180	0.080
$1/\eta _{u}^{[2]} $	SD	0.001	0.045	0.025	0.004
$1/\eta _{d}^{[2]} $	Mean	0.030	0.109	0.175	0.177
$1/\eta _{d}^{[2]} $	SD	0.009	0.068	0.027	0.062

Estimate	K	b	c	$\beta $	${{\sigma }_{1}}$	$\gamma $	${{H}^{[1]}} $	$\lambda _{u}^{[1]} $	$\lambda _{d}^{[1]} $	$1/\eta _{u}^{\left[ 1 \right]} $	$1/\eta _{d}^{[1]} $
Mean	25.14	-0.52	0.42	0.29	0.11	0.23	0.55	0.09	0.06	0.04	0.15
SD	3.27	0.03	0.02	0.01	0.00	0.00	0.00	0.00	0.02	0.00	0.01