Journal of Geographical Sciences >
Simulation on the stochastic evolution of hydraulic geometry relationships with the stochastic changing bankfull discharges in the Lower Yellow River
Song Xiaolong, e-mail: xlsong@tju.edu.cn |
Received date: 2019-02-28
Accepted date: 2019-09-12
Online published: 2020-07-25
Supported by
National Key Research and Development Program of China, No.2017YFC0404303()
Copyright
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.
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 . DOI: 10.1007/s11442-020-1758-z
Figure 1 The Gaocun-Sunkou reach of the Lower Yellow River |
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) | ||
---|---|---|---|---|---|---|---|---|
Discharge Qf (m3/s) | IS coefficient $\xi_{f}$(kg·s/m6) | |||||||
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 |
* The flood season is from July to October. |
Table 2 The estimated results of the unknown parameters set for the SDEs-Eq.(8a) |
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 3 The estimated results of the unknown parameters set for the SDEs-Eq.(8b) |
Estimate | Slope (S) | Width (B) | Depth (D) | Velocity (U) | |
---|---|---|---|---|---|
m | Mean | 0.184 | 0.771 | 0.560 | 0.424 |
SD | 0.038 | 0.218 | 0.092 | 0.068 | |
${{\sigma }_{2}}$ | Mean | 0.073 | 0.143 | 0.276 | 0.130 |
SD | 0.148 | 0.047 | 0.069 | 0.023 |
Table 4 The estimated results of the unknown parameters set for the jump-diffusion Eq. (10a) |
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 5 The estimated results of the unknown parameters set for the jump-diffusion Eq. (10b) |
Estimate | Slope (S) | Width (B) | Depth (D) | Velocity (U) | |
---|---|---|---|---|---|
m | Mean | -0.086 | 0.264 | 0.350 | 0.310 |
SD | 0.009 | 0.043 | 0.059 | 0.046 | |
${{\sigma }_{2}}$ | Mean | 0.075 | 0.301 | 0.300 | 0.160 |
SD | 0.014 | 0.065 | 0.065 | 0.055 | |
$\lambda _{u}^{[2]} $ | Mean | 0.180 | 0.300 | 0.311 | 0.394 |
SD | 0.001 | 0.014 | 0.029 | 0.065 | |
$\lambda _{d}^{[2]} $ | Mean | 0.180 | 0.300 | 0.327 | 0.426 |
SD | 0.004 | 0.026 | 0.054 | 0.025 | |
$1/\eta _{u}^{[2]} $ | Mean | 0.059 | 0.079 | 0.180 | 0.080 |
SD | 0.001 | 0.045 | 0.025 | 0.004 | |
$1/\eta _{d}^{[2]} $ | Mean | 0.030 | 0.109 | 0.175 | 0.177 |
SD | 0.009 | 0.068 | 0.027 | 0.062 |
Table 6 The estimated results of the unknown parameters set for the fractional jump-diffusion Eq. (13a) |
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 7 The estimated results of the unknown parameters set for the fractional jump-diffusion Eq. (13b) |
Estimate | Slope (S) | Width (B) | Depth (D) | Velocity (U) | |
---|---|---|---|---|---|
m | Mean | -0.141 | 0.704 | 0.350 | 0.880 |
SD | 0.011 | 0.043 | 0.026 | 0.063 | |
${{\sigma }_{2}}$ | Mean | 0.090 | 0.500 | 0.640 | 0.260 |
SD | 0.015 | 0.035 | 0.055 | 0.017 | |
${{H}^{[2]}} $ | Mean | 0.471 | 0.349 | 0.471 | 0.301 |
SD | 0.054 | 0.013 | 0.026 | 0.063 | |
$\lambda _{u}^{[2]} $ | Mean | 0.374 | 0.410 | 0.361 | 0.554 |
SD | 0.072 | 0.075 | 0.026 | 0.064 | |
$\lambda _{d}^{[2]} $ | Mean | 0.380 | 0.410 | 0.367 | 0.556 |
SD | 0.095 | 0.075 | 0.023 | 0.052 | |
$1/\eta _{u}^{[2]} $ | Mean | 0.088 | 0.488 | 0.300 | 0.230 |
SD | 0.041 | 0.048 | 0.011 | 0.033 | |
$1/\eta _{d}^{[2]} $ | Mean | 0.087 | 0.453 | 0.105 | 0.170 |
SD | 0.025 | 0.064 | 0.023 | 0.028 |
Figure 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 |
Figure 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 |
Figure 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 |
Figure 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 |
Figure 6 Comparison of stochastic average with measurements in the Gaocun-Sunkou reach of the Lower Yellow River under the three model conditions |
Figure 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) |
Table 8 The correlation coefficients of time-varying stochastic average Zw, $\zeta $,$\Omega $ with Q |
Correlation | Zw | $\zeta $ | $\Omega $ | Q |
---|---|---|---|---|
Q | 0.035 | 0.140 | 0.523 | 1 |
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