16.1.1 Stations Suide and Liujiahe

These two stations yield similar results as follows:

1. 
   

   a. The relation between discharge and suspended sediment yield is obviously nonlinear for the discharge lower than 100 cms and the relation tends to be linear in logarithm domain for the discharge higher than 100 cms. The fitted copula function properly follows this observed trend.

   
   
2. 
   

   b. The linear USGS regression equation in the logarithm domain (or the power function in real domain) tends to overestimate the suspended sediment yield for discharge higher than 100 cms at both locations.

   
   
3. 
   

   c. From the forecast results, it again shows that the copula-based approach provides a very reasonable forecast for sediment yield, especially in the case of high sediment yield due to high runoff events. During the flood season, one may directly use the copula approach to project the possible suspended sediment rushing downhill.

   
   
4. 
   

   d. For the same dataset for testing the forecast power, the USGS equation significantly overestimates sediment yield. The results obtained directly from the USGS equation are not reliable.

16.1.2 Stations Gaojiabao and Wangdaohengta

Like the stations at Suide and Liujiahe, similar results are obtained for Gaojiabao and Wangdaohengta stations as discussed in what follows:

1. 
   

   a. Unlike the Suide and Liujiahe stations, the scatterplots tend to be more linear at Gaojiabao and Wangdaohengta stations. As a result, both the copula and USGS equation approaches follow the trend reasonably well.

   
   
2. 
   

   b. The forecast performance using both approaches reach very similar results. The suspended sediment at Wangdaohengta is better forecasted than that at Gaojiabao (which exhibits as more of a nonlinear relation than that at Wangdaohengta).

Besides the obtaining the preceding results by grouping the stations with similar results together, we may gain more information from geomorphological and geological aspects:

1. 
   

   i. The slope of the river channel in different subreaches may be the main deciding factor for the relation between discharge and suspended sediment. The higher the percentage of slope (i.e., steeper than 1.5%), the more nonlinear the relation becomes. As the flattest station, Wangdaohengta reaches the most linear relation compared to all other stations (Figure 16.1; Wangdaohengta (A)). With the increase of slope, the nonlinearity becomes more and more obvious from low to high as Gaojiabao, Suide, and Liujiahe.

   
   
2. 
   

   ii. For the sections with steep river channels, the sediment from the upper subreach may be easier to be transported as suspended sediment. The local sediment may also be easier to be picked up and transported as suspended sediment. Suide station, located in the typical Loess region, may be a good example to carry both sediment from the upper subreach and local sediment as suspended sediment during flood events. Liujiahe station, located in the rock mountain region, may be a good example to carry sediment from upper subreach as suspended sediment.

   
   
3. 
   

   iii. For the sections with flat river channels, for the same amount of runoff, the flow velocity and the corresponding shear stress will be significantly reduced. As a result, the suspended sediment from the upper subreach may be deposited as a bed load rather than suspended sediment. Wangdaohengta station in the soft-rock region may be a good example to illustrate sediment deposition in a flat river channel and linear relation in the logarithm domain. Gaojiabao station in the sand region, which is slightly more nonlinear than Wangdaohengta station, may be a good example to explain the deposition with more suspended sediment transported than Wangdaohengta: (a) the overall slope at Gaojiabao is slightly steeper than that at Wangdaohengta and (b) the particle size of the underlying sand surface is generally smaller than that of soft-rock.

Above all, this case study suggests to apply the copula approach to construct the discharge-sediment rating curve for steeper channels, while the USGS regression equation in the logarithm domain may be safely applied for flatter channels.

Applying the sediment-rich middle reach of Yellow River with four major underlying surfaces, the case study indicates the following:

• 
  

  The USGS regression equation may work as well as the copula-based method when the channel is flat. In this case, sediment may be harder to be transported or it may actually move as bed load rather than suspended sediment due to low flow velocity and low shear force. For the flat channel, the type of underlying surface does not seem to be the dominating factor for the suspended sediment transport.

  
  
• 
  

  The copula-based method works much better than the USGS regression equation when the channel is steep. For the steep channel, the relation of discharge and suspended sediment in the logarithm domain is no longer linear. The nonlinearity for the discharge-suspended sediment rating curve is dependent on the underlying surface when the discharge is lower than a certain threshold. However, for the discharge higher than the certain threshold, the nonlinear relation seems to be replaced by the linear relation similar to the linear relation for the flatter channel.

16.2.1 Event-Based Sediment Dataset

Precipitation, discharge, and sediment yield are related. In this section, we are going to use the events collected from Flume #3 at Santa Rita experimental watershed (1975–2018) as the case study to further evaluate this interrelation. In the available dataset, sediment data were missing from 1990–2000. Figure 16.2 shows the watershed map retrieved from tucson.ars.ag.gov (USDA, n.d.). As shown in Figure 16.2, Flume 3 covers a small area of 6.81 acres. Table 16.4 lists the events selected from the available dataset.

Figure 16.2 Santa Rita experimental watershed (retrieved from tucson.ars.ag.gov).

16.2.2 Empirical Analysis of Sediment Dataset

Before we proceed with the copula application, we first perform an empirical analysis using the sediment dataset. The sample statistics listed in Table 16.5 show that the target variables are skewed to the right and tailed. As shown in Figure 16.3, the histogram and kernel density frequency yield the same results. It should be noted that the support of the kernel density needs to be positive based on the nature of dataset. Table 16.6 lists the rank-based Kendall and Spearman correlation coefficients of the pair variables. Results indicate the positive dependence for all pair variables. The scatter plots and chi- and K-plots further confirm the positive dependence structure for the pair variables, as shown in Figures 16.4 and 16.5.

Figure 16.3 Histogram and kernel density frequencies for the univariate variables.

Figure 16.4 Scatter plots of sediment yields versus rainfall depth, runoff volume, and peak runoff.

Figure 16.5 K- and chi-plots for the pair variables.

Now, with the dependence structure identified, we may further study the dependence with the use of copula theory.

16.2.3 Dependence Study of Runoff Volume and Sediment Yield with Copula Theory

We have shown that the pair variables are positively dependent. Additionally, there exists a higher degree of association between sediment yield versus runoff volume and sediment yield versus peak runoff than that between sediment yield versus rainfall depth. To illustrate the copula application in sediment analysis, we will first do bivariate analysis using sediment yield and runoff volume followed by the multivariate sediment analysis.

Before we apply the copula theory to the sediment yield and runoff volume, we will first compute the tail dependence coefficient. Applying all three empirical tail dependence coefficients LOG, SEC, and CFG, we compute the following: λ̂USEC=0.696,λ̂ULOG=0.683,λ̂UCFG=0.686. From the computed empirical upper-tail dependence coefficient, there is minimal difference in the results with the use of three approaches. Thus, we may apply the copula belonging to the extreme value family to do bivariate analysis. As discussed in Chapter 4, we will apply the Gumbel–Hougaard copula.

The empirical probabilities computed from the kernel density with positive support (Table 16.7) are applied to estimate the Gumbel–Hougaard copula parameter. Additionally, Figure 16.6 plots the comparison of kernel density–based CDF with that computed using the Weibull plotting-position formula. The scatter plots of CDFs (i.e., Figure 16.6(c)) visually indicates the upper-tail dependence between sediment yield and runoff volume. Applying the pseudo-MLE, the parameter is estimated as θ = 2.822θ=2.822. The corresponding theoretical upper-tail dependence coefficient is computed as λUGH=2−21θ=2−212.822=0.722, which is slightly higher than its empirical estimation. Applying the Rosenblatt goodness-of-fit test discussed in Chapters 3 and 4, we compute P=0.31,SnB=0.15. The results of the goodness-of-fit test further confirm that the Gumbel–Hougaard copula may be properly applied to do bivariate sediment analysis.

Figure 16.6 Comparison of kernel density with the Weibull plotting-position formula and scatter plots of empirical CDFs.

The plots in Figure 16.7 yield the same results in regard to the comparison of (i) empirical Kendall distribution versus parametric Kendall distribution for the fitted Gumbel–Hougaard copula, (ii) the empirical copula versus the fitted Gumbel–Hougaard copula, (iii) the empirical CDF (kernel density approach) versus simulated variates from the fitted Gumbel–Hougaard copula (frequency domain), and (iv) observations versus simulated variates (real domain).

Figure 16.7 Comparison of the fitted Gumbel–Hougaard copula with observations: (a) Kendall distribution, (b) copula, (c): variables in the frequency domain, (d) variables in real domain.

To this end, we have shown that a copula can successfully model the dependence of runoff volume and sediment yield. To better understand the interrelation among rainfall, runoff, and sediment yield, we will study multivariate dependence using rainfall depth, runoff volume, and sediment yield in the following section.

16.2.4 Multivariate Dependence Study of Rainfall Depth, Runoff Volume, and Sediment Yield

To study the multivariate dependence among rainfall depth, runoff volume, and sediment yield, we will apply two approaches: (i) meta-Student t copula and (ii) the vine copula with runoff volume as the center variable. To investigate the ability of using rainfall depth and runoff volume to predict sediment yield, we use the first 59 observations (Table 16.7) to build the dependence model and the last 10 observations to study the predictability of rainfall depth and runoff volume.

Dependence Model with the Meta-Student t Copula

Applying the meta-Student t copula discussed in Section 7.2.2, Table 16.8 lists the estimated parameter (i.e., the correlation matrix and degree of freedom) with pseudo-MLE in which the empirical CDF is again computed with the kernel density function approach. The application of kernel density function allows us to find the empirical CDF of “future” values continuously across ℝ⁺ℝ+ (listed in Table 16.8 for rainfall depth and runoff volume).

Table 16.8. Estimated parameter for the meta-Student t copula with pseudo-MLE.

	Sediment yield	Correlation matrix Rainfall depth	Runoff volume	Degree of freedom
Sediment yield	1	0.709	0.857
Rainfall depth	0.709	1	0.915	ν = 5.817ν=5.817
Runoff volume	0.857	0.915	1

We have shown in other chapters that the meta-elliptical copula (i.e., meta-Student t copula here) may be successfully applied to model the dependence structure, since the meta-elliptical copula is built on the transformed data (i.e., univariate meta-Student t transformation) as discussed in Chapter 7. Here, we will proceed to the forecast study for the sediment yield directly using the parameters listed in Table 16.8 and variates in Table 16.9.

Table 16.9. Univariate CDF computed for the last 10 observations using the kernel density function built with the first 59 observations.

Rainfall depth		Runoff volume
Inch	CDF	ft3	CDF
0.36	0.086	2,053	0.032
0.36	0.086	2,053	0.032
0.97	0.525	4,356	0.226
1.095	0.595	7,623	0.477
0.53	0.225	4,786	0.266
0.44	0.152	5,451	0.324
0.61	0.287	7,146	0.448
0.685	0.342	3,850	0.179
0.645	0.313	5,682	0.343
0.755	0.390	6,323	0.392

With rainfall depth and runoff volume known, the conditional distribution of sediment yield for the given rainfall depth and runoff volume may be written using Equation (7.56) as follows:

∑11=10.9150.9151,∑12=∑21T=0.7090.857,∑22=1,ν2∣1=7.817.

Applying the median forecast (i.e., P(sediment yield<=S|Rainfall depth = d, Runoff volume = v) = 0.5), we have TS−μs∣d,vΣs∣d,v0.5νS∣d,v=0.5, where S is the sediment yield forecasted. Using the same approach as that in Example 7.11, the forecasted sediment yields are plotted in Figure 16.8. Figure 16.8 also plots the forecasted sediment yields from bivariate sediment analysis. It is seen that the meta-Student t copula and GH copula (using the same parameter estimated for the entire dataset in Section 16.2.3) yield similar forecast results. Even though the observations fall into the 95% confidence interval constructed using the Student t copula, there exist visible differences among the observations and forecasts.

Figure 16.8 The forecasted sediment yields from trivariate and bivariate sediment analysis.

Dependence Model with Vine Copula

As in the preceding section, the first 59 observations are used to build the vine copula (shown in Figure 16.9), and the last 10 observations are applied for forecast purposes. The Gumbel–Hougaard copula is applied to model the runoff volume and sediment yield as well as rainfall depth and runoff volume of T1, and the Frank copula is applied for T2. The parameters estimated are also shown in Figure 16.9. It is noted that S, V, and D represent sediment yield, runoff volume, and rainfall depth, respectively. Comparisons shown in Figures 16.10 and 16.11 indicate the appropriateness of applying the fitted vine copula to do the trivariate sediment analysis. The Rosenblatt goodness-of-fit test also confirms that the fitted vine copula may properly model the dependence (SnB=0.059,P=0.635).

Figure 16.9 Vine structure and the parameter estimated.

Figure 16.10 Comparison of observations to the simulated variates in frequency domain.

Figure 16.11 Comparison of observations and simulated variates in real domain.

With the fitted vine copula, the median forecast of the sediment yield may be computed using the similar procedure for the second-order copula-based Markov process. In detail, the forecast of sediment yield is performed with the following steps:

1. 
   

   1. Compute the empirical CDF for the last 10 observations of rainfall depth (D) and runoff volumes (V), using the kernel density function fitted to the first 59 observed rainfall depth (listed in Table 16.9). Here we apply the assumption that variables are random and the first 59 observations may represent the population statistics.

   
   
2. 
   

   2. Compute conditional probability p1=PD≤dV=v=∂CGHFdFv3.589∂Fv.

   
   
3. 
   

   3. Compute the conditional probability p₂ = P(S ≤ s| V = v)p2=PS≤sV=v from P(S ≤ s| D = d, V = v) = 0.5PS≤sD=dV=v=0.5 as: p2=PS≤sV=v=P−1p10.5=CFrank−condtional−1p10.5−2.702. The conditional Frank copula is listed in Chapter 4.

   
   
4. 
   

   4. Compute the probability of the forecasted sediment yield by setting p₂ = P(S ≤ s| V = v)p2=PS≤sV=v and Fs=CGH−conditional−1Fvp22.955.

   
   
5. 
   

   5. Interpolate the forecasted sediment yield in the real domain with the use of the kernel density fitted to the first 59 observations.

Comparisons in Figure 16.12 indicate a very similar performance between the meta-Student t and vine copula.

Figure 16.12 Comparison of the forecasted sediment yield from vine, Student t, and Gumbel–Hougaard copulas with the observed sediment yield.