## Abstract

In the previous chapters, we have briefly introduced applications of copulas to analyses of rainfall, streamflow, drought, water quality, and compound extremes, as well as network design. In this chapter, we will introduce suspended sediment transport. Two case studies will be discussed to (i) apply copulas to construct the discharge-sediment rating curve using the Yellow River dataset; and (ii) investigate the dependence among precipitation, discharge, and sediment yield using the event-based dataset retrieved from the flume #3 at Santa Rita experimental watershed.

### 16.1 Discharge-Sediment Rating Curve Construction

In this section, application of copulas will be illustrated for modeling suspended sediment transport in regard to discharge-sediment rating curve by:

i. Identifying the study region and collecting discharge and suspended sediment data

ii. Applying the mixed copula that may reasonably capture the upper, lower, and overall dependence to model the discharge-suspended sediment data

iii. Comparing the performance of the copula approach to the classic USGS discharge-suspended sediment rating equations

iv. Interpreting the results according to the underlying surface and channel characteristics

The Yellow River, the most famous sediment-rich river system in the world, is used here as a case study for discharge-sediment rating curves. More specifically, sediment transport is studied for the middle reach of the Yellow River basin. The middle reach is the major sediment source area supplying 90% of the total sediment yield and 38% of the total water resources of the middle Yellow River (MYR) basin. Geologywise, the underlying surface materials change dramatically from north to south, varying from soft rock (i.e., Pisha Rock by locals), sand, loess, to rock mountains (Li and Li, 1994; Wang et al., 2007; Ni et al., 2008) with the median particle size decreasing from north (0.088 mm) to south (0.018 mm). To evaluate the study, we choose four representative stations for each geologically identified unique underlying surface: (1) typical loess – SuiDe, (2) soft-rock sand – Wangdaohengta, (3) rock mountain – Liujiahe, and (4) sand – Gaojiabao. Throughout the case study, we will apply both the classic USGS rating equation and copulas and also compare their performance. To prepare the dataset, the discharge values less than average discharge will be dropped out of the dataset, since we are more concerned with the large amount of suspended sediment transported during runoff events.

The discharge-sediment rating curve has been commonly applied to forecast suspended sediment yield or concentration. The classic USGS sediment rating curve (i.e., Equation (16.1)) through either power function or log-linear function has been commonly applied to achieve this end:

*SSL*=

*aQ*

^{b}

*or*ln (

*SSL*) =

*a*

^{∗}+

*b*ln (

*Q*) :

*a*

^{∗}= ln (

*a*)

In the previous chapters, we have discussed different types of copulas as well as their applications. In this section, we will apply mixed copula. Let *U* and *V* represent the marginals of discharge and suspended sediment, respectively; then we can write the mixed copula as follows:

*C*(

*u*,

*v*;

**) =**

*θ**w*

_{1}

*C*

_{1}(

*u*,

*v*;

*θ*_{1}) +

*w*

_{2}

*C*

_{2}(

*u*,

*v*;

*θ*_{2}) +

*w*

_{3}

*C*

_{3}(

*u*,

*v*;

*θ*_{3}); ∃

*w*

_{1}+

*w*

_{2}+

*w*

_{3}= 1

Commonly, copulas chosen are the copula with *λ*_{L}≠0λL≠0, the meta-Gaussian copula, and the copula with *λ*_{U}≠0λU≠0. Here we will choose the survival Gumbel–Hougaard copula, Gumbel–Hougaard copula, and Gaussian copula as the mixture. The survival Gumbel–Hougaard copula mirrors the Gumbel–Hougaard copula. As a result, the survival Gumbel–Hougaard copula has the lower-tail dependence. With the preceding mixture, Equation (16.2) may be rewritten as follows:

In Equation (16.3), the survival Gumbel–Hougaard copula (CSGH) can be expressed as follows:

Again, denoting *c*c as the copula density, the copula density of CSGH can be given as follows:

The lower- and upper-tail dependence coefficient is given for the survival Gumbel–Hougaard and Gumbel–Hougaard copula as follows:

Substituting Equation (16.3a) into Equation (16.3), the density function for the mixture copula can be given as follows:

*c*

_{mix}(

*u*,

*v*;

**) =**

*θ**w*

_{1}

*c*

^{GH}(

*u*,

*v*;

*θ*

_{1}) +

*w*

_{2}

*c*

^{Gaussian}(

*u*,

*v*;

*θ*

_{2}) +

*w*

_{3}

*c*

^{GH}(1 −

*u*, 1 −

*v*)

Similar to the discussions in other application chapters, to predict the suspended sediment, one needs to estimate the expected suspended discharge with *E*(*SSL*| *Q* = *q*)ESSLQ=q from

*P*(

*SSL*≤

*ssl*|

*Q*=

*q*) = 0.5

Equation (16.5) may be also called as the median forecast. Applying the copula theory, Equation (16.5) may be rewritten as follows:

*P*(

*SSL*≤

*ssl*|

*Q*=

*q*) =

*P*(

*F*

_{SSL}≤

*F*

_{SSL}(

*ssl*)|

*F*

_{Q}=

*F*

_{Q}(

*q*)) = 0.5

In Equation (16.5a), *P*(*F*_{SSL} ≤ *F*_{SSL}(*ssl*)| *F*_{Q} = *F*_{Q}(*q*))P(FSSL≤FSSLssl|FQ=FQq) can be written through copula as follows:

Let hFSSLFQθ=∂CFSSLFQθ∂FQFQ=FQq; we will have the following:

*F*

_{SSL}=

*h*

^{−1}(0.5,

*F*

_{Q}(

*q*),

**θ**)

The 90% bound may be written through VaR(5%) (i.e., lower bound) and VaR(95%) (i.e., upper bound) as follows:

With the computed FSSL,FSSL5%,FSSL95%, we usually compute the corresponding value in the real domain with the fitted parametric marginal distributions. In this case, we will use the kernel density function (with a normal kernel) with the positive support to estimate the projected sediment yield. Application of the kernel density function for prediction avoids the possible ill-identification of the marginal distributions due to the very high skewness. From the sample statistics listed in Table 16.1, it is seen that sediment yield is heavily skewed and tailed.

Station | Mean (kg/s) | Standard deviation (kg/s) | Skewness | Kurtosis |
---|---|---|---|---|

Suide | 11.262.25 | 32.037.33 | 7.54 | 86.83 |

Gaojiabao | 1.273.16 | 7.481.30 | 12.24 | 185.30 |

Wangdaoheng | 6.260.81 | 29.372.95 | 7.46 | 62.33 |

LiujiaHe | 11.698.75 | 46.025.54 | 9.28 | 115.36 |

Parameters of the mixture copula are estimated with the pseudo-MLE. The Weibull plotting-position formula is applied to compute the empirical marginals. In the case of model parsimony, the copula will be dropped out of the mixture if its corresponding weight is less than 0.1. For example, if the weights for both the Gaussian (*w*_{2})w2) and survival Gumbel–Hougaard copula (*w*_{3})w3) are less than 0.1, the mixture copula will be reduced to the Gumbel–Hougaard copula. With this procedure and the maximum likelihood applied to pseudo-observations (i.e., the empirical marginal of discharge and suspended sediment), Table 16.2 lists the estimated parameters of four stations from both copula approach as well as the USGS log-linear regression equations. The results of copula approach in Table 16.2 indicate that the Gumbel–Hougaard copula is the only copula needed based on the model selection procedure (i.e., only consider the copulas with the weight higher than 10% in the mixture). This is quite understandable due to the procedure of data processing: (1) omit the [discharge, sediment] pair when the discharge is lower than the average discharge; and (2) omit the [discharge, sediment] when the sediment yield is less than 0.5% of the average sediment yield.

Underlying | Geographic characteristics | Copula | USGS | ||||
---|---|---|---|---|---|---|---|

Surface | Station | Area (km^{2}) |
Slope | Gaussian | GH | SGH | [a^{∗}, b]a∗b |

Loess | Suide | 3893 | 22.99^{a} |
– | 3.071 | – | [1.388, 2.254] |

Sand | Gaojiabao | 2095 | 0.74 | – | 1.863 | – | [–2.373, 2.822] |

Soft-rock | Wangdaohengta | 3390 | 0.29 | – | 2.955 | – | [–0.023, 2.134] |

Rock mountain | Liujiahe | 2361 | 44.25^{a} |
– | 2.544 | – | [–0.629, 2.595] |

*Note:* ^{a} The percentage of the slope steeper than 1.5%.

To visually compare the copula approach with the USGS equation, Figure 16.1 compares the fitted copula function and USGS equation (in Figure 16.1A) as well as their forecast power (in Figure 16.1B). The forecast results (Figure 16.1B) are listed in Table 16.3.

Q (cms)^{a} |
SSY(kg/s)^{a,b} |
Copula^{c} |
USGS^{c} |
Q (cms) | SSY(kg/s) | Copula | USGS |
---|---|---|---|---|---|---|---|

Suide station | Liujiahe station | ||||||

97.7 | 86,073.7 | 76,957.20 | 122,621.87 | 128 | 110,976 | 89,958.95 | 156,852.25 |

127 | 87,249 | 112,057.81 | 221,483.23 | 184 | 114,632 | 131,095.07 | 402,279.65 |

104 | 92,040 | 84,049.35 | 141,170.48 | 196 | 13,1908 | 138,174.95 | 473,956.12 |

138 | 96,462 | 125,758.45 | 267,092.44 | 147 | 133,182 | 106,072.00 | 224,639.69 |

113 | 101,022 | 94,727.58 | 170,214.45 | 210 | 144,060 | 145,989.60 | 566,893.19 |

164 | 103,812 | 158,278.90 | 394,136.30 | 192 | 158,976 | 135,863.41 | 449,260.01 |

164 | 103,812 | 158,278.90 | 394,136.30 | 326 | 167,890 | 269,808.11 | 1774,991.57 |

164 | 104,960 | 158,278.90 | 394,136.30 | 254 | 182,626 | 176,613.58 | 928,772.23 |

115 | 109,020 | 971,64.83 | 177,081.01 | 297 | 184,140 | 229,288.33 | 1,393,760.25 |

169 | 111,033 | 164,714.59 | 421,742.65 | 241 | 185,811 | 165,330.82 | 810,388.83 |

138 | 112,056 | 125,758.45 | 267,092.44 | 275 | 209,000 | 200,007.38 | 1,141,415.85 |

176 | 121,968 | 173,833.17 | 462,146.64 | 282 | 210,090 | 208,958.40 | 1,218,358.53 |

158 | 124,346 | 150,666.33 | 362,375.14 | 279 | 210,924 | 205,069.38 | 1,185,005.20 |

158 | 124,346 | 150,666.33 | 362,375.14 | 264 | 215,952 | 187,018.04 | 1,026,674.31 |

178 | 139,018 | 176,451.53 | 474,069.33 | 252 | 221,004 | 174,722.80 | 909,911.49 |

198 | 146,916 | 202,558.00 | 602,680.75 | 322 | 224,112 | 264,266.82 | 1,719,020.60 |

204 | 147,900 | 210,140.26 | 644,633.46 | 407 | 231,990 | 385,953.85 | 3,157,341.74 |

204 | 157,896 | 210,140.26 | 644,633.46 | 272 | 236,096 | 196,331.14 | 1,109,380.45 |

231 | 200,046 | 241,611.05 | 853,096.43 | 343 | 242,158 | 293,295.70 | 2,025,308.37 |

251 | 213,099 | 262,330.06 | 1,028,698.61 | 438 | 244,842 | 433,235.20 | 3,819,957.26 |

309 | 224,952 | 328,976.59 | 1,643,641.81 | 375 | 304,125 | 338,053.86 | 2,552,848.84 |

Gaojiabao Station | Wangdaohengta Station | ||||||

23.9 | 13,503.5 | 6,01.10 | 723.16 | 51.1 | 3,980.69 | 3,729.80 | 4,313.90 |

36.1 | 14,692.7 | 2,573.53 | 2315.61 | 36.5 | 4,526 | 2,102.76 | 2,104.27 |

42.3 | 15,397.2 | 3,527.72 | 3621.66 | 43.1 | 4,870.3 | 2,825.67 | 2,999.91 |

53.5 | 15,408 | 4,535.20 | 7027.18 | 30.5 | 5,215.5 | 1,503.63 | 1,434.50 |

38.3 | 16,200.9 | 2,973.66 | 2736.31 | 66 | 6,217.2 | 5,577.68 | 7,446.53 |

57.9 | 18,296.4 | 5,916.65 | 8783.01 | 31.9 | 7,081.8 | 1,636.34 | 1,578.65 |

60.3 | 18,873.9 | 7,124.40 | 9849.59 | 120 | 7,992 | 22,696.57 | 2,6662.30 |

51.5 | 19,415.5 | 4,171.17 | 6310.86 | 84.8 | 11,108.8 | 9,176.62 | 12,711.38 |

51.6 | 20,691.6 | 4,185.74 | 6345.50 | 50.4 | 11,491.2 | 3,650.05 | 4,188.80 |

54.1 | 22,289.2 | 4,674.29 | 7251.86 | 51 | 11,985 | 3,718.39 | 4,295.91 |

70.3 | 22,917.8 | 11,992.07 | 15186.77 | 138 | 12,806.4 | 32,135.92 | 35,925.11 |

72.9 | 23,182.2 | 14,130.16 | 16825.71 | 97.4 | 19,967 | 13,019.05 | 17,082.53 |

88.4 | 24,663.6 | 21,679.14 | 28989.32 | 74.4 | 25,072.8 | 6,926.16 | 9,615.22 |

57.9 | 25,012.8 | 5,916.65 | 8783.01 | 87.6 | 25,579.2 | 9,919.53 | 13,623.63 |

52.9 | 26,502.9 | 4,409.92 | 6807.06 | 182 | 27,482 | 59,230.34 | 64,838.33 |

59.4 | 29,581.2 | 6,633.95 | 9440.36 | 146 | 28,032 | 36,673.52 | 40,514.79 |

73 | 34,310 | 14,219.50 | 16890.92 | 121 | 29,403 | 23,187.85 | 27,138.58 |

70.8 | 40,780.8 | 12,360.18 | 15493.56 | 134 | 36,448 | 29,939.41 | 33,739.90 |

89.7 | 423,38.4 | 22,507.14 | 30208.52 | 174 | 45,240 | 53,886.25 | 58,908.88 |

98.5 | 46,886 | 30,084.53 | 39338.98 | 89.6 | 57,881.6 | 10,489.26 | 14,295.85 |

108 | 50,220 | 37,758.66 | 51011.10 | 122 | 63,440 | 23,683.93 | 27,619.34 |

*Notes:* ^{a} Observed; ^{b}: SSY stands for suspended sediment yield; ^{c} forecast from discharge (Q) in column 1.