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

Xiaolong SONG; Deyu ZHONG; Guangqian WANG

doi:10.1007/s11442-020-1758-z

Journals >Journal of Geographical Sciences >Volume 30 >Issue 5 >Page 843 > Article

Journal of Geographical Sciences
Vol. 30, Issue 5, 843 (2020)

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

Xiaolong SONG^1、2, Deyu ZHONG^1、*, and Guangqian WANG¹

Author Affiliations

¹State Key Laboratory of Hydro-science and Engineering, Tsinghua University, Beijing 100084, China

²State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, China

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DOI: 10.1007/s11442-020-1758-z Cite this Article

Xiaolong SONG, Deyu ZHONG, Guangqian WANG. 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 Copy Citation Text

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Fig. 1. The Gaocun-Sunkou reach of the Lower Yellow River

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Comparison of the calculated and measured values of bankfull channel geometries in the Gaocun-Sunkou reach of the Lower Yellow River under the Model-1 condition

Fig. 2. Comparison of the calculated and measured values of bankfull channel geometries in the Gaocun-Sunkou reach of the Lower Yellow River under the Model-1 condition

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Fig. 3. Comparison of the calculated and measured values of bankfull channel geometries in the Gaocun-Sunkou reach of the Lower Yellow River under the Model-2 condition

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Fig. 4. Comparison of the calculated and measured values of bankfull channel geometries in the Gaocun-Sunkou reach of the Lower Yellow River under the Model-3 condition

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Fig. 5. The time-varying process of effective probabilistic stability thickness of hydraulic geometry in the Gaocun-Sunkou reach of the Lower Yellow River under the three model conditions

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Fig. 6. Comparison of stochastic average with measurements in the Gaocun-Sunkou reach of the Lower Yellow River under the three model conditions

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Fig. 7. The time-varying probability distribution of riverbed stability indices, hydraulic width/depth ratio and stream power in the Gaocun-Sunkou reach of the Lower Yellow River based on Fractional Jump-Diffusion model (13)

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Year	Flood season’s average value (*)			Bankfull discharge Q (m³/s)	Slope S (‰)	Width B (m)	Depth D (m)	Velocity U (m/s)
Year	DischargeQ_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.

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

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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

Table 2.

The estimated results of the unknown parameters set for the SDEs-Eq.(8a)

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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

Table 3.

The estimated results of the unknown parameters set for the SDEs-Eq.(8b)

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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

Table 4.

The estimated results of the unknown parameters set for the jump-diffusion Eq. (10a)

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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

Table 5.

The estimated results of the unknown parameters set for the jump-diffusion Eq. (10b)

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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

Table 6.

The estimated results of the unknown parameters set for the fractional jump-diffusion Eq. (13a)

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Estimate		Slope (S)	Width (B)	Depth (D)	Velocity (U)
m	Mean	-0.141	0.704	0.350	0.880
m	SD	0.011	0.043	0.026	0.063
${{\sigma }_{2}}$	Mean	0.090	0.500	0.640	0.260
${{\sigma }_{2}}$	SD	0.015	0.035	0.055	0.017
${{H}^{[2]}} $	Mean	0.471	0.349	0.471	0.301
${{H}^{[2]}} $	SD	0.054	0.013	0.026	0.063
$\lambda _{u}^{[2]} $	Mean	0.374	0.410	0.361	0.554
$\lambda _{u}^{[2]} $	SD	0.072	0.075	0.026	0.064
$\lambda _{d}^{[2]} $	Mean	0.380	0.410	0.367	0.556
$\lambda _{d}^{[2]} $	SD	0.095	0.075	0.023	0.052
$1/\eta _{u}^{[2]} $	Mean	0.088	0.488	0.300	0.230
$1/\eta _{u}^{[2]} $	SD	0.041	0.048	0.011	0.033
$1/\eta _{d}^{[2]} $	Mean	0.087	0.453	0.105	0.170
$1/\eta _{d}^{[2]} $	SD	0.025	0.064	0.023	0.028