# Estimation of Suspended Sediment Loads Using Copula Functions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{2}, and the Kuzlovec basin itself 0.71 km

^{2}. The basic characteristics of the Kuzlovec torrent are given in Table 1.

#### 2.2. Data

^{3}and the maximum SSC value was 2364 g/m

^{3}[4]. Moreover, the frequency analyses of the SSC data indicate that the SSC value corresponding to the 10-year return period is about 1810 g/m

^{3}[4]. Hellmann rain gauge was used to measure accumulated daily rainfall at the Cankova station at 7 a.m. In total, 281 events were used to additionally test the copula model for estimating the SSL values. Detailed description of the methodology used to determine individual events is given in the next section. All this data (discharge, SSL, rainfall) was measured by the Slovenian Environment Agency (ARSO) and the data is also publically available [45].

#### 2.3. An Event-Based Sample Selection Methodology

^{3}/s), accumulated rainfall (P) (mm), and suspended sediment load (SSL) (kg or t).

#### 2.4. Copula Function

**X**= (X

_{1}, …, X

_{d}) is a random vector, F its joint cumulative distribution function (CDF), and F

_{1}, …, F

_{d}, marginal CDFs, then by Sklar’s theorem [52] there exists a d-copula

**C**:[0, 1]

^{d}→ [0, 1] such that:

_{i}: = F(X

_{i}) are uniformly distributed on [0, 1]. Moreover, if all CDFs are also strictly increasing, we obtain:

_{1}, …, F

_{d}, Equation (1) defines a multivariate CDF F with marginals F

_{1}, …, F

_{d}.

_{1}, α

_{2}, α

_{3}$\text{}\in \left[0,1\right]\text{}$are shape parameters [59,60]. The following combinations were tested in this study: (a) ${C}_{1}\text{}\mathrm{and}\text{}{C}_{2}$ are independence copulas (this model has 3 parameters: α

_{1}, α

_{2}, α

_{3}); (b) ${C}_{1}\text{}\mathrm{and}\text{}{C}_{2}$ are independence copula and trivariate symmetric Archimedean copula (this model has 4 parameters: α

_{1}, α

_{2}, α

_{3}and$\text{}{\theta}_{1}$); (c) ${C}_{1}\text{}\mathrm{and}\text{}{C}_{2}$ are trivariate symmetric Archimedean copulas (this model has 5 parameters: α

_{1}, α

_{2}, α

_{3}$\mathrm{and}\text{}{\theta}_{1}\text{}\mathrm{and}\text{}{\theta}_{2}$). This means that for case (a) 1 combination; (b) 8 combinations; (c) 16 combinations (all possible combinations of Clayton, Frank, Joe and Gumbel-Hougaard copulas were used) for the function in Equation (5) were tested.

_{n}, which was defined by [63]. The R copula package was used for copula parameter estimation and the goodness-of-fit test presented in this paper [64]. The most suitable copula function among Frank, Clayton, Joe, Gumbel-Hougaard and Khoudraji-Liebscher copula was selected using the model selection criterion (function xvCopula) that is based on the k-fold cross-validation and it is implemented in the program R copula package [64]. The detailed description of the criterion can be found in [65].

_{3}(U

_{3}= F

_{SSL}(SSL)) given the values of U

_{1}(U

_{1}= F

_{Q}(Q)) and U

_{2}(U

_{2}= F

_{P}(P)) can be shown to be (Appendix A):

- For each measured (known) pair of variables Q and P 10,000 uniform random variables [0, 1] were generated;
- For each of the 10,000 uniform randomly generated variables, Equation (7) was solved numerically using the Newton’s method for solving nonlinear equations [72];
- For each of the solutions of Equation (7), the inverse Probability Integral Transform (PIT) ($SSL\text{}=\text{}{F}_{SSL}^{\left[-1\right]}\left({u}_{3}\right)$) was used to transform the solution from the copula space [0, 1] to the real space and consequently estimate the SSL value.
- For each known pair of variables Q and P, which corresponds to the specific event, a sample of 10,000 possible SSL values was obtained.
- The median value of all 10,000 possible SSL values was selected as the estimated SSL value and 50% confidence intervals for each event were also determined. Alternatively, the mode could be selected as the most likely value in some other cases.

#### 2.5. Regression Models and Performance Criteria

^{2}) and residual analysis (graphical presentation and descriptive statistics of residuals) were used for comparison of different methods (Copula, MLR and EXP).

## 3. Results and Discussion

#### 3.1. Kuzlovec Torrent

#### 3.1.1. Estimation of Copula Model Parameters for the Kuzlovec Torrent

^{−7}), and 0.70 (z-value: 4.4; p-value: 0.0001), respectively. Therefore, the null hypothesis was rejected with selected significance level of 0.05. The calculated correlation coefficients indicate that the dependence between pairs of variables is similar. Moreover, the exchangeability test results for pairs of variables P-Q, P-SSL and Q-SSL were 0.022 (p-value: 0.99), 0.022 (p-value: 0.98) and 0.021 (p-value: 0.72), respectively. This indicates that the selection of symmetric copula functions (e.g., Clayton, Frank, Gumbel-Hougaard, Joe), which have one parameter to model the dependence among three variables, seems reasonable for the Kuzlovec case study.

#### 3.1.2. Comparison with Other SSL Estimation Techniques and Estimation of SSL Values Based on Measured Q and P

#### 3.2. Gornja Radgona Station on the Mura River

#### 3.2.1. Estimation of Copula Model Parameters for the Gornja Radgona Station

^{−13}), 0.64 (z-value: 15.9; p-value: 2 × 10

^{−16}) and 0.28 (z-value: 6.9; p-value: 4 × 10

^{−12}), respectively. These correlation coefficients do not depend on the transformation of the data. Moreover, the exchangeability test results for pairs of variables Q-P, Q-SSL, and P-SSL were 0.04 (p-value: 0.24), 0.01 (p-value: 0.94) and 0.04 (p-value: 0.15), respectively. Since the dependence between pairs of variables is not the same, the symmetric copula function with one parameter is not the most suitable model. Therefore, the Khoudraji-Liebscher copula function (Equation (5)) was used to model the data from the Gornja Radgona station on the Mura River. Different combinations of the Khoudraji-Liebscher copula were tested (independence copula and Joe, Clayton, Frank and Gumbel-Houaard copulas were used as C

_{1}and C

_{2}in Equation (5)). The adequacy of different combinations ((a), (b) and (c) defined in Section 2.4) was tested using the Cramér-von Mises test S

_{n}. Based on the selected significance level of 0.05 the following combinations of C

_{1}and C

_{2}(Equation (5)) could not be rejected: (i) Joe-C

_{1}and Joe-C

_{2}(statistic: 0.10; p-value: 0.07); (ii) Gumbel-Hougaard-C

_{1}and Gumbel-Hougaard-C

_{2}(statistic: 0.08; p-value: 0.10) and (iii) Gumbel-Hougaard-C

_{1}and Joe-C

_{2}(statistic: 0.07; p-value: 0.14). In total 25 combinations were tested and all other combinations were rejected at the selected significance level of 0.05 ((a), (b) and (c) defined in Section 2.4). Moreover, to select the most suitable copula function the copula leave-one-out cross-validation selection criterion was used [65]. The criterion results were 162.1, 164.8 and 157 for the combinations (i), (ii) and (iii), respectively. Based on these results the combination of Gumbel-Hougaard-C

_{1}and Gumbel-Hougaard-C

_{2}was used for the estimation of the SSL values for the Gornja Radgona station. The selected model has 5 parameters. In some other cases different combination of C

_{1}and C

_{2}could be selected in order to reduce the number of parameters (over-parametrization issue). For example, the application of independence copula (case (a) defined in Section 2.4) leads to a model with 3 parameters, which is positive from the over-parametrization perspective. However, in this case study models with smaller number of parameters (cases (a) and (b) defined in Section 2.4) were rejected by the selected goodness-of-fit test for copula functions. The copula parameters were estimated using the maximum pseudo-likelihood method. The estimated parameters ${\theta}_{1}\text{}\mathrm{and}\text{}{\theta}_{2}$ for the Gumbel-Hougaard (C

_{1}) and Gumbel-Hougaard (C

_{2}) copulas were 1.78 and 3.12, respectively (where U

_{1}= F

_{Q}(Q), U

_{2}= F

_{SSL}(SSL) and U

_{3}= F

_{P}(P) according to the notations used in Equation (5)). Moreover, the shape parameters α

_{1}, α

_{2}and α

_{3}were 0.60, 0.63 and 2 × 10

^{−5}, respectively. Different parametric distributions such as Gumbel, Generalized Pareto, Pearson type 3, log-Pearson type 3 and GEV were tested but none of them gave an adequate fit to the data. Therefore, the nonparametric distribution function, which is defined with Equation (6), was used to model the selected variables (Q, P, and SSL). This distribution function was tested using the Kolmogorov-Smirnov test and for all variables the nonparametric distribution could not be rejected at the chosen significance level of 0.05.

#### 3.2.2. Comparison with Other SSL Estimation Techniques

## 4. Conclusions

- The proposed copula model for estimating the SSL values based on the measured Q and P values yielded meaningful results. According to some performance criteria and graphical presentation of the results the copula model gives comparable results to those obtained using other tested models (MLR and EXP). For the Gornja Radgona station the copula model yielded better fit to the actual measured SSL values than other tested methods. In this case study 281 events were available to estimate the copula model parameters and nonparametric distributions were selected as marginal distributions. However, for the Kuzlovec torrent much smaller number of events was analyzed and parametric distribution functions were used. The differences in the estimation results could also be a consequence of different copulas that were selected (symmetric and Khoudraji-Liebscher copulas). Using the copula model the probabilistic estimation of the SSL values can be obtained, which is not possible using other tested methods. Moreover, the smallest residual values were characteristic of the estimation procedure that was carried out using copula function, which indicates an important advantage of the proposed copula method compared to other tested methods. However, there were some differences between the low-medium and medium-high magnitude SSL events.
- The proposed copula based model is flexible. Both symmetric and Khoudraji-Liebscher copula functions were used to construct the copula model based on the dependence characteristics of the analyzed variables. Furthermore, other copula functions with more parameters and different properties such as Gaussian copula or Vine copulas could be used in this model to estimate the SSL values based on the Q and P. Similarly, also different marginal distribution functions can be selected, even nonparametric. The proposed copula model where nonparametric marginal distribution functions were used is more robust tool that is not significantly affected by transformations of the marginal data.
- An event-based copula model used in this study could easily be upgraded with additional variables (e.g., bedload, water electrical conductivity measurements, antecedent sediment transport conditions or antecedent soil moisture), because copula functions of higher dimensions can be constructed relatively easily. Moreover, similar model could also be used for the estimation of different environmental variables (e.g., biogeochemical model-water chemistry).
- Unlike some other techniques, the presented event-based model also captures the sediment lag effect. In the future it would be reasonable to consider also the sediment depletion (exhaustion) effect (e.g., antecedent sediment transport), which can have a considerable impact on the SSL values during consecutive events.
- The proposed event-based copula model can be a useful tool for estimating sediment budgets. The methodology was successfully applied to two different case studies, a small forested torrent and a larger river catchment and comparable results were obtained in both cases (in first case 20-min data was used while in the second case daily data was used).

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

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**Figure 1.**Location of the Gradaščica River basin and Gornja Radgona station on the topographic map of Slovenia with the main stream network.

**Figure 2.**Graphical presentation of an event-based definition of variables used in the proposed copula model.

**Figure 3.**Autocorrelation plots for the data from the Kuzlovec torrent (upper plots) and Mura River (lower plots) where for the Q series for the Mura case the transformed data are showed.

**Figure 4.**Graphical presentation of 21 events that were used to define the copula model for the Kuzlovec torrent. Crosses indicate multiple-peaks events and circles indicate single-peak events.

**Figure 5.**Distribution of possible SSL pseudo-values (solutions of Equation (7)), estimated SSL values in the copula space, 50% confidence intervals and measured SSL values for four events, which happened in different seasons for the Kuzlovec torrent.

**Figure 6.**Diagnostic plots showing residuals versus fitted values for three tested models (Copula, EXP and MLR) for the Kuzlovec torrent.

**Figure 7.**Graphical presentation of 281 events that were used to define the copula model for the Gornja Radgona station on the Mura River.

**Figure 8.**Diagnostic plots showing residuals versus fitted values for three tested models (Copula, EXP and MLR) for the Gornja Radgona station on the Mura River where for the EXP and MLR models the original data is used.

**Table 1.**Basic characteristics of the Kuzlovec torrent and Gornja Radgona station on the Mura River.

Name | Kuzlovec | Gornja Radgona |
---|---|---|

Basin area (km^{2}) | 0.71 | 10,197 |

Basin elevation (minimum; maximum; mean) (m a.s.l.) | 394; 847; 631 | 203; 3075; ~1015 |

Mean basin slope (%) | 52 | ~25 |

Mean channel slope (%) | 22 | ~0.7 |

Main channel length (km) | 1.3 | ~300 |

Mean annual precipitation (mm) | 1600–1800 | 950 |

Statistic/Variable | P (mm) | Q (L/s) | SSL (kg) |
---|---|---|---|

Min | 1.6 | 6.7 | 0.4 |

1st quartile | 16.8 | 9.9 | 3.9 |

Median | 23.0 | 15.6 | 17.2 |

Mean | 27.2 | 31.1 | 94.3 |

3rd quartile | 37.8 | 45.8 | 167.9 |

Max | 63.0 | 125.9 | 470.3 |

**Table 3.**Performance criteria results and summary statistics for the estimated SSL values for the Kuzlovec torren.

Statistic/Model | Copula | MLR | EXP |
---|---|---|---|

MAE (kg) | 40.3 | 34.76 | 42.5 |

RMSE (kg) | 68.3 | 59.42 | 61.7 |

NSE | 0.74 | 0.80 | 0.79 |

R^{2} | 0.77 | 0.80 | 0.79 |

Min SSL (kg) | 1 | −16.1 | 14.7 |

Median SSL (kg) | 13.3 | 36.47 | 40.6 |

Max SSL (kg) | 451 | 476.5 | 494.0 |

**Table 4.**Basic properties of the SSL, Q and P values for different seasons for the Kuzlovec torrent.

Season | Mean P (mm) | Max Q (L/s) | SSL (kg) | SSL_{0.25} (kg) | SSL_{0.75} (kg) |
---|---|---|---|---|---|

Summer 2013 | 11.4 | 36.0 | 99 | 58 | 172 |

Autumn 2013 | 17.8 | 125.9 | 1429 | 805 | 2787 |

Winter 2013–2014 | 26.8 | 138.6 | 1724 | 989 | 3293 |

Spring 2014 | 12.1 | 43.5 | 360 | 217 | 645 |

**Table 5.**Summary statistics of the defined variables for the data from the Gornja Radgona station on the Mura River.

Statistic/Variable | P (mm) | Q (m^{3}/s) | SSL (t) |
---|---|---|---|

Min | 30.1 | 62.7 | 357.7 |

1st quartile | 37.7 | 169 | 3453 |

Median | 49.3 | 244.3 | 9317 |

Mean | 58.3 | 295.1 | 26,410 |

3rd quartile | 68.4 | 374 | 26,500 |

Max | 186.9 | 1237 | 303,000 |

**Table 6.**Performance criteria results and summary statistics for the estimated SSL values for the Gornja Radgona station on the Mura River.

Statistic/Model | Copula | MLR | EXP |
---|---|---|---|

MAE (t) | 12,403 | 15,500 | 13,637 |

RMSE (t) | 26,261 | 26,230 | 25,400 |

NSE | 0.65 | 0.65 | 0.67 |

R^{2} | 0.66 | 0.65 | 0.67 |

Min SSL (t) | 900 | −20,020 | 1410 |

Median SSL (t) | 8768 | 16,280 | 16,210 |

Max SSL (t) | 283,500 | 214,500 | 297,900 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bezak, N.; Rusjan, S.; Kramar Fijavž, M.; Mikoš, M.; Šraj, M.
Estimation of Suspended Sediment Loads Using Copula Functions. *Water* **2017**, *9*, 628.
https://doi.org/10.3390/w9080628

**AMA Style**

Bezak N, Rusjan S, Kramar Fijavž M, Mikoš M, Šraj M.
Estimation of Suspended Sediment Loads Using Copula Functions. *Water*. 2017; 9(8):628.
https://doi.org/10.3390/w9080628

**Chicago/Turabian Style**

Bezak, Nejc, Simon Rusjan, Marjeta Kramar Fijavž, Matjaž Mikoš, and Mojca Šraj.
2017. "Estimation of Suspended Sediment Loads Using Copula Functions" *Water* 9, no. 8: 628.
https://doi.org/10.3390/w9080628