2. Study Area and Database
2.1. Study Area
2.2. Data and Data Preprocessing
3.1. Sediment Rating Curve (SRC)
3.2. Optimization Tecnhiques
3.2.1. Conventional Correction Factors
Food and Agriculture Organization (FAO)
Non-Parametric Smearing Estimator (CF2)
3.2.2. Metaheuristic Algorithms
Genetic Algorithm (GA)
- Developing a set of initial random answers; these answers, which are the primary solutions to the problem, are called chromosomes and each one is made up of sets of genes. In the present study, the coefficients a and b in the SRC model are considered as genes and form a chromosome.
- Comparing, ranking, and selecting the best chromosomes; after developing the initial population of chromosomes, to determine its suitability, the efficiency of each chromosome in estimating the suspended sediment must be determined. At this point, using Equation (2) and the values of genes in each chromosome (a and b coefficients), the amount of suspended sediment for the training data is estimated. Then, the suitability of that chromosome using the objective function (root mean squared error (RMSE)) is determined as :
- Selecting pairs (parents) for reproduction; at this stage, using selection operators, a pair of chromosomes from the set of primary chromosomes in the previous stage are determined as the parents of the next generation. To accomplish this, the widely used roulette wheel selection method was applied . In fact, in this method, chromosomes with more favorable answers are more likely to be selected.
- Crossover; the production of new and better chromosomes is accomplished to further investigate the solution space (space containing possible coefficients for the SRC model). In this study, the blending method was employed to combine genes in the parent chromosomes and perform the reproduction. In each generation, the number of reproductions was determined by a parameter called the crossover rate.
- Mutation; mutation is a mechanism that leads to a completely random change in the genes of chromosomes (answers to the problem). This prevents the early convergence and getting stuck in local minima, enabling a better search within the answers space.
- Convergence; convergence implies that the GA, by repeating successive generations, is no longer able to find better answers to the problem. There are various ways to stop the genetic algorithm, e.g., the number of repetitions of generations reaching a certain level of error and lack of significant progress in error reduction.
Particle Swarm Optimization (PSO)
- Generation of the initial random population with random positions and velocities, each called a particle (a and b coefficients in the SRC model are assumed to be equivalent to one particle).
- Evaluation of the cost or fitness of each particle; at this stage, the amount of suspended sediment for the training dataset is estimated using Equation (2) and the values for each particle (a and b coefficients). Then, their suitability is evaluated using the objective function (Equation (7)).
- Recording the best position for each particle (pbest) and the best position among all particles (gbest); at this step, each particle moves at a speed that can be adjusted to the search space and retains the best previous position in its memory. In addition, in the total search space, the best gained position by the group is shared with all particles. Each particle in an assumed space is shown as a position and velocity vector. The position of each particle is obtained by comparison between the current position and the best value it has achieved (pbest). Moreover, the best response that each particle has attained so far from the pbest is identified as gbest;
- Updating the position and velocity vector of all particles; in this step, the transition of the particles to new positions is evaluated. In addition, the velocity and position of each particle are corrected by Equations (8) and (9), respectively.
- Convergence test; this algorithm is repeated for a predetermined number of generations or it is executed until the problem converges to an optimal solution.
Imperialist Competitive Algorithm (ICA)
- Generating the random initial countries (a and b coefficients in the SRC model are assumed equivalent to one country).
- Dividing the countries into two categories based on the objective function of the problem (Equation (7)). Countries with the lowest amounts of objective function are assumed as imperialist and the rest are colonies.
- Determining the number of colonies of each imperialist; to this aim, the power of each imperialist must be evaluated. It is obvious that the stronger the imperialist, the greater the number of its colonies.
- Applying the assimilation policy after the formation of the initial empires; in this algorithm, the assimilation policy is modeled as the movement of colonies towards imperialists.
- Revolution in countries can be considered as a sudden and accidental change in the situation of the colonized countries.
- Comparing the colonies and imperialists (intra-group competition); sometimes a colony, by moving towards an imperialist, reaches a new situation in which it has a lower cost function than the imperialist. In this case, the colony and the imperialist change positions.
- Evaluation of empires (intergroup competition); at this stage, a colony is removed from a weaker empire and transferred to another empire. If the empire has no colony, its imperialist is transferred as a colony to another empire. As a result, during colonial competition, the power of larger empires gradually increases, and weaker empires will be eliminated.
- Finally, continuing the algorithm until the termination condition is observed. The end limit of colonial competition is when we have a single empire in the world with colonies that are very close to the imperialist country in terms of situation.
3.3. Data Separation Techniques
- Seasonal: The measured data for SSC were classified into spring, summer, autumn, and winter ;
- Discharge classes: Data were divided based on annual average discharge such that in the first category discharge was less than average discharge; in the second category, discharge was ≥the average, but less than twice the average; in the third category, discharge was ≥twice the average ;
- High water and low water periods: Mean monthly discharge was compared to the mean annual discharge. The months in which mean discharge was ≥ mean annual discharge were considered as the high water period and the months in which the mean discharge was less than the mean annual were considered as the low water period ;
- Hydrograph state: The daily hydrograph of each water year was plotted and data were classified into three series based on rising and falling limbs or base flow of the hydrograph . Moreover, to assess the effect of these groups on the efficiency of models in estimating suspended sediment, results were compared with a group without data separation (group 5).
3.4. Machine Learning (ML) Model
3.5. Model Evaluation and Comparison
4. Results and Discussion
4.1. Results of the SRC Model Based on Data Separation and Non-Separation
4.2. Results of Optimization of the SRC Using Classical Methods and Metaheuristic Algorithms
4.3. Results of SVR Models with Data Separation and Non-Separation
4.4. Determination of the Best Method of Data Separation
4.5. The Most Effective Model for Estimating SSC
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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|Study Period||Statistical Parameter|
|High water period||Training||0.07||25.71||3.1||1.95||1.3||215|
|Low water period||Training||0||5.41||1.31||1||2.01||265|
|Study Period||Statistical Parameter|
|High water period||Training||0.14||15,152.21||611.06||2028.42||6.35||215|
|Low water period||Training||0.01||8231.67||264.22||1257.81||5.91||265|
|Without any separation (a)||Entire period||35.04||1.86|
|Discharge Classes (c)||15.92||2.8|
|High water-low water periods (d)||High water period||25.46||2.18|
|Low water period||35.21||1.69|
|Hydrograph State (e)||Falling limb||60.58||1.54|
|Model||Study Period||RMSE |
|without any separation||Entire period||1366.96||0.19||292.23||0.35|
|High water/low water periods||High water period||1189.56||0.27||287.07||0.36|
|Low water period||1151.44||0.33||265.54||0.41|
|Hydrograph State||Falling limb||1310.15||0.13||399.12||0.24|
|Model Name||Equation||RMSE |
|without any separation||Entire period||2.5||1||0.1|
|High water-low water periods||High water period||2||2.5||0.1|
|Low water period||0.1||1||0.01|
|Hydrograph State||Falling limb||0.1||1.5||0.1|
|Model||Study Period||RMSE |
|without any separation||Entire period||1069.89||0.50||201.09||0.52|
|High water—low water periods||High water period||1106.5||0.51||201.63||0.53|
|Low water period||929.72||0.53||189.78||0.55|
|Hydrograph State||Falling limb||1119.2||0.38||237.3||0.43|
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