# Superiority of Hybrid Soft Computing Models in Daily Suspended Sediment Estimation in Highly Dynamic Rivers

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{t-1}) is the most crucial input variable for daily SSC estimation of the Koyna River basin.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Data Collection

^{2}on the Deccan plateau. The study area comes under the survey of India toposheets 47G/9, 47G/11, 47G/11, 47G/13, 47G/14, 47G/15, 47G/16, 47K/13, and 47K/14. The study area comes under varying climatic and topographic conditions. The annual rainfall at the upstream part of the basin is 5000 mm, reducing to 866 mm downstream. About 88% rainfall occurs in the monsoon (1 June to 30 September) season. In this study, the mean areal rainfall of the study area was determined using the Thiessen polygon method in ArcGIS 10.2 software. The study area comes under a subtropical climate. The winter season starts in October and extends up to January, while the summer season extends from February to May. The daily mean monthly maximum temperature varies between 31 °C to 37 °C, while the daily mean monthly minimum temperature varies between 10 °C to 14 °C. Agriculture is the primary source of income for people who lives in the River basin. The soil at the upstream part of the basin is light laterite, while the central and downstream area is under black cotton soil. The elevation in the basin varies between 534 m to 1437 m above the mean sea level. The dominant part of the basin is under the steep sloping condition with varying topography and prone to soil loss. The entire basin area is covered by agriculture (717.13 km

^{2}), bare soil (491.69 km

^{2}), open forest (342.60 km

^{2}), dense forest (159.84 km

^{2}), built-up land (120.20 km

^{2}), and water body (85.68 km

^{2}), as shown in Figure 1b which was prepared using ArcGIS 10.2 software in this study runoff and sediment measurements were conducted at the basin’s outlet, situated at Warunji village in the Satara district.

#### 2.2. ANN

_{i}) with their corresponding connection weights (W

_{i}) plus the bias or threshold value (b) of a neuron. Usually, the net function is in the linear form given as:

_{i}is an input variable, W

_{i}is the connection weight from the i

^{th}neuron in the input layer, and b is the bias/threshold value of the neuron [39]. The net function (u) at a hidden node is transformed into output (y) using a non-linear activation function. More details of ANN were added in the Appendix A.

#### 2.3. ANFIS

_{i}and B

_{i}are Membership Functions (MFs) for input x and y, respectively. Simultaneously, p

_{i}, q

_{i,}and r

_{i}are the design (consequent) parameters estimated during the training process. The TSK model’s fuzzy reasoning mechanism derives an output (f) with inputs (x and y). General ANFIS structure with two inputs, two rules, and a single output. The functioning of the ANFIS structure (five layers) is described below. The description of the ANFIS layers functions was presented in the Appendix B.

#### 2.4. Subtractive Clustering

_{1}, ….., x

_{n}), subtractive clustering assumes each data point act as a candidate for representing the cluster center. The subtractive clustering depends on data density. The density index at any point x

_{i}is expressed as:

_{i}represents density index, and r

_{a}is positive (r

_{a}> 0), indicating the neighborhood radius in each cluster center. So, the data point which has more neighborhood points indicates more potential to represent as cluster center. Those data points which are located outside the radius create less impact on the density index. Selection of clustering radius receives greater importance while determining the number of clusters. The high value of r

_{a}causes the minimum number of clusters and vice versa. After determining a data point with a high potential to act as a cluster center, say X

_{c1}is a point act as a first cluster center determined by the D

_{ci}density index. The expression recalculates the next density index for each data point Xi:

_{b}is a positive constant (r

_{b}> 0), which indicates the neighborhood radius for which the most significant reduction in the density will be achieved. To prevent closely spaced cluster centers, r

_{b}is usually equal to 1.5 times of r

_{a}. After this density measurement, the next cluster center X

_{c2}is selected. The same process is repeated until sufficient numbers of cluster centers are achieved.

#### 2.5. Wavelet Transform

_{0}= 2 and n

_{0}= 1, where n

_{0}is the location parameter. It must be greater than zero and m

_{0}displays fined dilation step that is greater than one. The DWT scale and position are based on the power of two (dyadic scales and positions); this power of two logarithmic scalings of the translation and dilation is known as the dyadic grid arrangement. The dyadic wavelet function is defined as [41]:

_{i}, the dyadic wavelet transform becomes [26]:

_{a,b}is the wavelet coefficient for the discrete wavelet with scale m = 2

^{a}and location n = 2

^{a}, b. (i = 0, 1, 2,…, L-1; and L is an integer power of 2: L = 2

^{A}).Also, the signal’s smoothed component, which represents the time series’ overall trend, is considered T. The discrete inverse transform can reconstruct the signal x

_{i}as [42]:

_{a,b}(t) are details sub-signals at levels a = 1, 2,..., A and time dimension of t (t = 1, 2,.., b). The wavelet coefficients, T

_{a,b}(t) with (a = 1, 2,…, A), give the detailed sub-signals which can capture small features of interpretational value available in the time series data. The residual term T(t) indicates an approximate sub-signal, representing background information available in the time series data. An approximate sub-signal represents the general trend of the original time series signal. In contrast, a detailed sub-signal represents high-frequency components of the original time series signal [42]. Using these approximate and detail sub-signals, the characteristics of time series like jump, period, hidden period, and dependence can be identified easily [22]. When the original signal passes through low and high pass filter at each decomposition level, it gets resolved into approximate and detail sub-signals; the decomposition at each level is satisfied by the condition given as:

#### 2.6. Mother Wavelets

#### 2.7. Gamma Test (GT)

^{N}denotes input, corresponding scaler c ϵ R denotes output. Here, the assumption is that the input vector consists of useful information that can affect output c. The relationship among variables in the system is assumed in the form as:

_{ratio}. To apply the GT, win Gamma

^{TM}software was used.

#### 2.8. Data Normalization

_{max}and S

_{min}are the maximum and minimum values of original time series data, n is the number of data points, and x is the original input variable’s normalized value.

#### 2.9. Training and Testing of Developed Models

^{n}number of fuzzy rules (here, n is the number of input variables, and m is the number of MFs per input). Hence, when the inputs increase slightly, fuzzy rules increase rapidly [43]. Grid partitioning can be easily used for solving problems with less than 6 input variables [44]. Hence, in this study, subtractive clustering was employed to develop ANFIS/WANFIS models. In total, eight input MFs were used by changing the number of MFs per input from 2 to 4. The rule base was constructed with OR logical operation by changing the number of rules from 2 to 4. A total of two output MFs, such as constant and linear, were used. Hence, individually 48 (8 input MFs × 2 output MFs × 3 rules) ANFIS models were developed by applying simple ANFIS or any WANFIS models technique by keeping the error tolerance of 0.001 and maximum iterations of 1000 in MATLAB (R2015a) software with a hybrid learning algorithm. In this study, different ANFIS/WANFIS models were developed using ‘’anfisedit’’ tool in MATLAB (R2015a) software with hybrid learning algorithm and Takagi–Sugeno–Kang (TSK) FIS.

#### 2.10. Performance Evaluation of Developed Models

#### 2.10.1. Quantitative Evaluation

#### 2.10.2. Hydrological Indices

#### Coefficient of Efficiency (CE)

#### Pooled Average Relative Error (PARE)

#### 2.11. Qualitative Evaluation

#### 2.12. Uncertainty Analysis

#### 2.12.1. Width of Uncertainty Band of Error Prediction (W_{e})

#### 2.12.2. 95% Confidence Interval of Error Prediction (CI_{e})

## 3. Results

#### 3.1. Data Analysis

#### 3.2. GT for Input Selection

_{t}and Q

_{t}are the present day’s rainfall and runoff, respectively. R

_{t}

_{− 1}, R

_{t}

_{− 2,}and R

_{t}

_{− 3}are the previous one, two- and three-days rainfall, respectively, Q

_{t}

_{− 1}, Q

_{t}

_{− 2,}and Q

_{t}

_{− 3}are the previous one, two- and three-days runoff, respectively, S

_{t}

_{− 1}, S

_{t}

_{− 2,}and S

_{t}

_{− 3}is the previous one, two- and three-days SSC, respectively. Here, initially, a total of 11 input variables (j) were selected for GT. Based on these 11 input variables, a total of 2

_{j}

_{− 1}(i.e., 2047) input combinations can be possible, but, in this study, reliable 58 input combinations were made and analyzed as presented in Table 3.

_{ratio}value. As per GT, the model (M14) with six input variables (R

_{t}, R

_{t}

_{− 1}, R

_{t}

_{− 2}, R

_{t}

_{− 3}, Q

_{t}, S

_{t}

_{− 1}) were selected and used to develop different models.

#### 3.3. Hydrological Model Development

#### 3.3.1. ANN/ANFIS Models for Daily SSC Prediction

#### 3.3.2. WANN/WANFIS Models for Daily SSC Prediction

#### 3.4. Quantitative Performance Evaluation of Developed Models for Daily SSC Prediction

^{2}= 0.74, and RMSE =3601.1 ton/day) at ANN structure 4-1-1. After analyzing the quantitative prediction performance among all 48 simple ANFIS models, it was revealed that RMSE, r, WI, CE, and PARE varies between 0.033 g/L to 0.044 g/L and 0.080 g/L to 1.23 g/L, 0.77 to 0.88 and −0.43 to 0.78, 0.71 to 0.80 and 0.27 to 0.74, 0.587 to 0.772 and −107.9 to 0.545, −1 × 10

^{−8}% to 2.3 × 10

^{−8}% and −0.736% to 0.975% during training and testing, respectively. Among all 48 simple ANFIS models, the ANFIS-29 model (triangular input MF, constant output MF and 3 MFs/input) performed better during the training and testing period. Similarly, for all 48 Haar-WANFIS models, it was found that RMSE, r, WI, CE, and PARE ranges from 0.016 g/L to 0.050 g/L and 0.074 g/L to 0.288 g/L, 0.69 to 0.97 and 0.1 to 0.82, 0.63 to 0.88 and 0.5 to 0.76, 0.47 to 0.947 and −4.955 to 0.605, −4 × 10

^{−9}% to 2.5 × 10

^{−9}% and −0.105% to 0.094% during training and testing, respectively. Here, the Haar-WANFIS-27 model (triangular input MF, linear output MF, and 4 MFs/input) performed better during both the training and testing period. After comparing the prediction performance of all 48 db2-WANFIS models, it was observed that RMSE, r, WI, CE, and PARE varies between 0.012 g/L to 0.087 g/L and 0.068 g/L to 0.808 g/L, 0.11 to 0.98 and −0.51 to 0.84, 0.00 to 0.91 and 0.24 to 0.77, −0.607 to 0.969 and −45.92 to 0.666, −5 × 10

^{−9}% to 0.029% and −0.96% to 0.115% during training and testing, respectively. Here, the db2-WANFIS-25 model (triangular input MF, linear output MF and 2 MFs/input) performed better during both the training and testing period. Similarly, among all 48 coif2-WANFIS models, it was revealed that RMSE, r, WI, CE, and PARE ranges between 0.013 g/L to 0.045 g/L and 0.06 g/L to 0.392 g/L, 0.76 to 0.98 and −0.40 to 0.87, 0.69 to 0.92 and 0.41 to 0.80, 0.573 to 0.964 and −10.04 to 0.745, −5 × 10

^{−9}% to 8.9 × 10

^{−8}% and −0.398% to 0.07% during training and testing, respectively. Among all 48 coif2-WANFIS models, the coif2-WANFIS-43 model (z input MF, linear output MF and 2 MFs/input) performed better for both the training and testing period. Our results are in line with Rajaee, et al. [48], who found high linear relationships from R2 of 0.72 to 0.87 and RMSE ranged from 1805.3 ton/day to 2459.6 ton/day applying ANFIS. While the R2 was from 0.62 to 0.67, and RMSE varied from 2543 ton/day to 2838 ton/day by using WANFIS.

#### 3.5. Qualitative Performance Evaluation of Developed Models for Daily SSC Prediction

#### 3.6. Uncertainty Analysis

#### 3.7. Sensitivity Analysis

_{t − 1}), the selected ANN-3model’s predictive performance decreased dramatically. Therefore, it is concluded that S

_{t − 1}is the most critical hydrologic variable in the daily SSC prediction. The order of sensitivity for daily SSC prediction was observed to be S

_{t − 1}followed by R

_{t − 3}, R

_{t}, R

_{t − 2}, Q

_{t}, R

_{t − 1}.

## 4. Discussion

#### Comparison of ANN, WANN, ANFIS, and WANFIS Models for Daily SSC Prediction

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

**Layer 1:**Every node in this layer creates a membership grade for an input variable. Every node i in this layer is a square node (or adaptive node), whose node function is defined as:

^{th}node, A

_{i}(or B

_{i}−2) is the fuzzy set associated with this node, and it is characterized by MFs shape. The MF may be any appropriate functions (continuous and piecewise differentiable) like Gaussian, Trapezoidal, Generalized bell, and Triangular shaped functions. Different researchers apply different MFs for the search for the solution to any problem. Considering, generalized bell function, the output of the first layer (i

^{th}node) is determined as:

_{i}, b

_{i}, c

_{i}) is the premise parameter set, and by varying this premise parameter, the shape of MF can change. For a Gaussian function, the output of the first layer (i

^{th}node) is determined as:

_{i}or σ

_{i}) is the premise parameter set, and by varying this premise parameter, the shape of MF can change. Here, MFs center is represented by c while MFs width is represented by σ.

**Layer 2:**Every node in this layer is a fixed node labeled as II, whose output multiplicate all incoming signals. The output O

_{i}

^{2}indicates the firing strength of a rule, and it can be determined as:

**Layer 3:**Every node in this layer is a fixed node labeled as N. The i

^{th}node in this layer determines the normalized firing strengths as:

**Layer 4:**Every node i in this layer is a square node with a node function given as:

_{i}, q

_{i}, r

_{i}} is the parameter set of this node (consequent parameters).

**Layer 5:**The single node in this layer is a fixed node labeled as sigma that computes the overall output as the summation of all incoming signals,

_{i}, q

_{i}, r

_{i}} and premise parameters {a

_{i}, b

_{i}, c

_{i}} are required to be optimized. Consequent parameters are identified using the least square method during the forward pass of the hybrid learning approach when the node outputs move forward. The error signals are propagated backward during the backward pass. The premise parameters are adjusted by using the gradient descent method [22].

## Appendix C

_{T[i,k]}represents kth (1 ≤ k ≤ p) nearest neighbor, in terms of Euclidean distance to b

_{i}(1 ≤ I ≤ N). GT derived from Delta function of the input vector is given as:

_{ratio,}can be used. It ranges from 0 to 1, and it is defined as:

^{2}(c) represents the variance of output (c). The value of V

_{ratio}closer to 1 represents a higher degree of predictability.

## Appendix D

#### Appendix D.1. Root Mean Squared Error (RMSE)

_{o}is the observed value, Q

_{p}is the predicted value, and n is the total number of values.

#### Appendix D.2. The Correlation Coefficient (r)

#### Appendix D.3. Willmott Index (WI)

^{2}) and Nash-Sutcliffe efficiency. WI lies between 0 to 1; the higher WI values indicate that predicted values show better agreement than observed values. Legates and McCabe Jr. [51] introduced a modified WI followed by a generic form of WI proposed by Willmott [52] to overcome the limitations of original WI against extreme values, which is expressed as:

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**Figure 1.**(

**a**) The geographical location and the gauging stations (

**b**) land use/land cover of the Koyna River basin.

**Figure 3.**Flowchart of the methodology adopted for daily suspended sediment concentration (SSC) estimation.

**Figure 5.**Schematic representation of wavelet coupled ANN/wavelet coupled ANFIS models for daily SSC prediction.

**Figure 6.**Decomposition of original rainfall time series using different mother wavelets. (

**a**) Haar, (

**b**) db2, (

**c**) coif2.

**Figure 7.**Decomposition of original runoff time series using different mother wavelets. (

**a**) Haar, (

**b**) db2, (

**c**) coif2.

**Figure 8.**Decomposition of original SSC time series using different mother wavelets. (

**a**) Haar, (

**b**) db2, (

**c**) coif2.

Statistical Parameters | Whole Data | Training Data Set | Testing Data Set | ||||||
---|---|---|---|---|---|---|---|---|---|

R_{t} (mm) | Q_{t} (m^{3}/s) | SSC_{t} (g/L) | R_{t} (mm) | Q_{t} (m^{3}/s) | SSC_{t} (g/L) | R_{t} (mm) | Q_{t} (m^{3}/s) | SSC_{t} (g/L) | |

Mean | 14.60 | 215.65 | 0.059 | 12.00 | 141.03 | 0.048 | 20.61 | 388.02 | 0.084 |

Standard Deviation | 22.91 | 405.67 | 0.088 | 17.33 | 191.73 | 0.069 | 31.53 | 646.81 | 0.118 |

Kurtosis | 11.85 | 41.33 | 20.03 | 7.62 | 14.54 | 17.105 | 6.81 | 15.67 | 13.261 |

Skewness | 3.01 | 5.38 | 3.643 | 2.51 | 3.41 | 3.175 | 2.47 | 3.48 | 3.180 |

Range | 193.32 | 4641 | 0.841 | 119.51 | 1397.00 | 0.717 | 193.32 | 4640.50 | 0.839 |

Minimum | 0.00 | 0.00 | 0.000 | 0.00 | 0.00 | 0.000 | 0.00 | 0.50 | 0.002 |

Maximum | 193.32 | 4641 | 0.841 | 119.51 | 1397.00 | 0.717 | 193.32 | 4641.00 | 0.841 |

Count | 854 | 854 | 854 | 596 | 596 | 596 | 258 | 258 | 258 |

Variable | S_{t} | R_{t} | R_{t-1} | R_{t-2} | R_{t-3} | Q_{t} | Q_{t-1} | Q_{t-2} | Q_{t-3} | S_{t-1} | S_{t-2} | S_{t-3} | p-Values |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

S_{t} | 1.00 | - | |||||||||||

R_{t} | 0.70 | 1.00 | 5.65 × 10^{−26} | ||||||||||

R_{t − 1} | 0.68 | 0.74 | 1.00 | 0.73 | |||||||||

R_{t − 2} | 0.61 | 0.58 | 0.74 | 1.00 | 0.97 | ||||||||

R_{t − 3} | 0.51 | 0.45 | 0.58 | 0.74 | 1.00 | 0.27 | |||||||

Q_{t} | 0.62 | 0.66 | 0.68 | 0.63 | 0.56 | 1.00 | 0.04 | ||||||

Q_{t − 1} | 0.54 | 0.48 | 0.66 | 0.68 | 0.63 | 0.89 | 1.00 | 0.78 | |||||

Q_{t − 2} | 0.44 | 0.37 | 0.48 | 0.66 | 0.68 | 0.75 | 0.89 | 1.00 | 0.93 | ||||

Q_{t − 3} | 0.35 | 0.29 | 0.37 | 0.48 | 0.66 | 0.65 | 0.75 | 0.89 | 1.00 | 0.10 | |||

S_{t − 1} | 0.72 | 0.51 | 0.70 | 0.68 | 0.61 | 0.59 | 0.62 | 0.54 | 0.44 | 1.00 | 1.82 × 10^{−27} | ||

S_{t − 2} | 0.60 | 0.41 | 0.51 | 0.70 | 0.68 | 0.56 | 0.59 | 0.62 | 0.54 | 0.72 | 1.00 | 0.00016 | |

S_{t − 3} | 0.49 | 0.33 | 0.41 | 0.51 | 0.70 | 0.52 | 0.56 | 0.59 | 0.62 | 0.60 | 0.72 | 1.00 | 0.18 |

Model | Model Input Combination | Mask | Gamma | V − Ratio |
---|---|---|---|---|

M1 | R_{t} | 10000000000 | 0.0047802000 | 0.5471600 |

M2 | R_{t}, R_{t − 1} | 11000000000 | 0.0042625000 | 0.4879000 |

M3 | R_{t}, R _{t− 1}, R_{t−2} | 11100000000 | 0.0041758000 | 0.4779800 |

M4 | R_{t}, R_{t − 1}, R_{t − 2}, R_{t − 3} | 11110000000 | 0.0032813667 | 0.4219459 |

M5 | R_{t}, R_{t − 1}, R_{t − 2}, Q_{t} | 11110000000 | 0.0038298000 | 0.4383700 |

M6 | R_{t}, R_{t − 1}, R_{t − 2}, R_{t − 3,} Q_{t} | 11111000000 | 0.0030169776 | 0.3879486 |

M7 | R_{t}, R_{t − 1}, R_{t − 2}, R_{t − 3,} Q_{t,} Q_{t − 1} | 11111100000 | 0.0034506204 | 0.4437100 |

M8 | R_{t}, R_{t − 1}, R_{t − 2}, R_{t − 3,} Q_{t,} Q_{t − 1,} Q_{t − 2} | 11111110000 | 0.0034489278 | 0.4434924 |

M9 | R_{t}, R_{t − 1}, R_{t − 2}, R_{t − 3,} Q_{t,} Q_{t − 1,} Q_{t − 2,} Q_{t − 3} | 11111111000 | 0.0032920565 | 0.4233205 |

M10 | R_{t}, R_{t − 1}, R_{t − 2}, R_{t − 3,} Q_{t,} Q_{t − 1,} Q_{t − 2,} Q_{t − 3,} S_{t − 1} | 11111111100 | 0.0032920565 | 0.4233205 |

M11 | R_{t}, R_{t − 1}, R_{t − 2}, R_{t − 3,} Q_{t,} Q_{t − 1,} Q_{t − 2,} Q_{t − 3,} S_{t − 1,} S_{t − 2} | 11111111110 | 0.0032920306 | 0.4233172 |

M12 | R_{t}, R_{t − 1}, R_{t − 2}, R_{t − 3,} Q_{t,} Q_{t − 1,} Q_{t − 2,} Q_{t − 3,} S_{t − 1,} S_{t − 2,} S_{t − 3} | 11111111111 | 0.0032920306 | 0.4233172 |

M13 | R_{t}, R_{t − 1}, R_{t − 2}, Q_{t,} S_{t − 1,} S_{t − 2} | 11101000110 | 0.0034097067 | 0.4384490 |

M14 | R_{t}, R_{t − 1}, R_{t − 2}, R_{t − 3,} Q_{t,} S_{t − 1} | 11111000100 | 0.0030159990 | 0.3878227 |

M15 | R_{t}, R_{t − 1}, R_{t − 2}, R_{t − 3,} Q_{t,} S_{t − 1,} S_{t − 2} | 11111000110 | 0.0030159998 | 0.3878228 |

M16 | R_{t}, R_{t − 1}, R_{t − 2}, R_{t − 3,} Q_{t,} S_{t − 1,} S_{t − 2,} S_{t − 3} | 11111000111 | 0.0030165903 | 0.3878988 |

M17 | R_{t}, R_{t − 1}, R_{t − 2}, R_{t − 3,} Q_{t,} Q_{t − 1,} S_{t − 1,} S_{t − 2,} S_{t − 3} | 11111100111 | 0.0034493755 | 0.4435500 |

M18 | R_{t}, R_{t − 1}, R_{t − 2}, R_{t − 3,} Q_{t,} Q_{t − 1,} Q_{t − 2,} S_{t − 1,} S_{t − 2,} S_{t − 3} | 11111110111 | 0.0034489289 | 0.4434925 |

M19 | R_{t}, R_{t − 1}, R_{t − 2}, Q_{t,} Q_{t − 1,} Q_{t − 2,} Q_{t − 3,} S_{t − 1,} S_{t − 2,} S_{t − 3} | 11101111111 | 0.0033026471 | 0.4246824 |

M20 | R_{t}, R_{t − 1}, R_{t − 2}, R_{t − 3,} Q_{t,} Q_{t − 1,} Q_{t − 2,} Q_{t − 3,} S_{t − 1,} S_{t − 2} | 11111111110 | 0.0032920306 | 0.4233172 |

M21 | R_{t}, R_{t − 1}, R_{t − 2}, R_{t − 3,} Q_{t,} Q_{t − 1,} Q_{t − 2,} S_{t − 1,} S_{t − 2,} S_{t − 3} | 11111110111 | 0.0034489289 | 0.4434925 |

M22 | R_{t}, R_{t − 1}, R_{t − 2}, Q_{t}, Q_{t − 1} | 11101100000 | 0.0042706000 | 0.4888370 |

M23 | R_{t}, R_{t − 1}, R_{t − 2}, R_{t − 3}, Q_{t}, Q_{t − 1} | 11111100000 | 0.0034506204 | 0.4437100 |

M24 | R_{t}, R_{t − 1}, R_{t − 2}, Q_{t}, Q_{t − 1}, Q_{t − 2} | 11111100000 | 0.0040742000 | 0.4663550 |

M25 | R_{t}, R_{t − 1}, R_{t − 2}, Q_{t}, Q_{t − 1}, Q_{t − 2}, S_{t − 1} | 11101110100 | 0.0040742000 | 0.4663550 |

M26 | R_{t}, R_{t − 1}, R_{t − 2}, Q_{t}, Q_{t − 1}, Q_{t − 2}, S_{t − 1}, S_{t − 2} | 11101110110 | 0.0040742000 | 0.4663550 |

M27 | R_{t}, R_{t − 1}, R_{t − 2}, S_{t − 2} | 11100000010 | 0.0041747000 | 0.4778580 |

M28 | R_{t}, R_{t − 1}, R_{t − 2}, S_{t − 1}, S_{t − 2} | 11100000110 | 0.0041719000 | 0.4775350 |

M29 | R_{t}, R_{t − 1}, R_{t − 2}, Q_{t}, S_{t − 1} | 11101000100 | 0.0038292000 | 0.4383100 |

M30 | R_{t}, R_{t − 1}, R_{t − 2}, Q_{t}, S_{t − 1}, S_{t − 2} | 11101000110 | 0.0038289000 | 0.4382700 |

M31 | R_{t}, R_{t − 1}, R_{t − 2}, Q_{t}, Q_{t − 1}, S_{t − 1}, S_{t − 2} | 11111100110 | 0.0042695000 | 0.4887090 |

M32 | R_{t − 1} | 01000000000 | 0.0045396000 | 0.5196200 |

M33 | R_{t − 1}, R_{t − 2} | 01100000000 | 0.0046157000 | 0.5283400 |

M34 | R_{t − 1}, R_{t − 2}, Q_{t} | 01101000000 | 0.0048902000 | 0.5597500 |

M35 | R_{t − 1}, R_{t − 2}, Q_{t}, S_{t − 1} | 01101000100 | 0.0048889000 | 0.5596100 |

M36 | R_{t − 1}, R_{t − 2}, Q_{t}, S_{t − 1}, S_{t − 2} | 01101000110 | 0.0048885000 | 0.5595600 |

M37 | R_{t − 1}, R_{t − 2}, Q_{t}, Q_{t − 1}, S_{t − 1}, S_{t − 2} | 01101100110 | 0.0045635000 | 0.5223630 |

M38 | R_{t − 1}, R_{t − 2}, Q_{t}, Q_{t − 1}, Q_{t − 2}, S_{t − 1}, S_{t − 2} | 01101110110 | 0.0044611000 | 0.5106410 |

M39 | R_{t − 2} | 00100000000 | 0.0059649000 | 0.6827700 |

M40 | R_{t − 2}, Q_{t} | 00101000000 | 0.0049465000 | 0.5662000 |

M41 | R_{t − 2}, Q_{t}, S_{t − 1} | 00101000100 | 0.0049457000 | 0.5661100 |

M42 | R_{t − 2}, Q_{t}, S_{t − 1}, S_{t − 2} | 00101000110 | 0.0049452000 | 0.5660450 |

M43 | Q_{t} | 00001000000 | 0.0053284000 | 0.6099170 |

M44 | Q_{t}, S_{t − 1} | 00001000100 | 0.0053036000 | 0.6070760 |

M45 | Q_{t}, S_{t − 1}, S_{t − 2} | 00001000110 | 0.0052017000 | 0.5954070 |

M46 | S_{t − 1} | 00000000100 | 0.0045528000 | 0.5211380 |

M47 | S_{t − 1}, S_{t − 2} | 00000000110 | 0.0046496000 | 0.5322160 |

M48 | S_{t − 2} | 00000000010 | 0.0060325000 | 0.6905120 |

M49 | R_{t}, R_{t − 2}, Q_{t}, S_{t − 1}, S_{t − 2} | 10101000110 | 0.0041118000 | 0.4706530 |

M50 | R_{t}, Q_{t}, S_{t − 1}, S_{t − 2} | 10001000110 | 0.0042217000 | 0.4832370 |

M51 | R_{t}, S_{t − 1}, S_{t − 2} | 10000000110 | 0.0039567000 | 0.4529040 |

M52 | R_{t}, S_{t − 2} | 10000000010 | 0.0042196000 | 0.4830030 |

M53 | R_{t}, R_{t − 1}, Q_{t}, S_{t − 1}, S_{t − 2} | 11001000110 | 0.0038594000 | 0.4417640 |

M54 | R_{t}, R_{t − 1}, S_{t − 1}, S_{t − 2} | 11000000110 | 0.0042129000 | 0.4822330 |

M55 | R_{t}, R_{t − 1}, S_{t − 2} | 11000000010 | 0.0042181000 | 0.4828270 |

M56 | R_{t}, R_{t − 1}, R_{t − 2}, S_{t − 1}, S_{t − 2} | 11100000110 | 0.0041719000 | 0.4775350 |

M57 | R_{t}, R_{t − 1}, R_{t − 2}, S_{t − 2} | 11100000010 | 0.0041747000 | 0.4778580 |

M58 | R_{t}, R_{t − 1}, R_{t − 2}, Q_{t}, S_{t − 2} | 11101000010 | 0.0038293000 | 0.4383140 |

**Table 4.**Quantitative performance evaluation indices of the best selected Artificial Neural Network (ANN) and wavelet coupled ANN (WANN) models.

Model | Architecture | Training | Testing | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

RMSE (g/L) | r | WI | CE | PARE (%) | RMSE (g/L) | r | WI | CE | PARE (%) | ||

ANN-3 | 6-3-1 | 0.040 | 0.81 | 0.78 | 0.655 | 0.0017 | 0.078 | 0.75 | 0.73 | 0.560 | −0.013 |

Haar-WANN-21 | 18-21-1 | 0.042 | 0.80 | 0.73 | 0.633 | 0.0041 | 0.069 | 0.83 | 0.74 | 0.661 | 0.006 |

db2-WANN-21 | 18-21-1 | 0.031 | 0.90 | 0.72 | 0.801 | 0.0236 | 0.061 | 0.86 | 0.75 | 0.734 | −0.012 |

coif2-WANN-11 | 18-11-1 | 0.021 | 0.95 | 0.81 | 0.903 | 0.0134 | 0.065 | 0.84 | 0.76 | 0.696 | −0.017 |

Model | Training | Testing | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

RMSE (g/L) | R | WI | CE | PARE (%) | RMSE (g/L) | r | WI | CE | PARE (%) | |

ANFIS-29 | 0.041 | 0.80 | 0.78 | 0.638 | 6.6 × 10^{−9} | 0.080 | 0.78 | 0.73 | 0.545 | 0.060 |

Haar-WANFIS-27 | 0.032 | 0.88 | 0.83 | 0.782 | 1.7 × 10^{−9} | 0.074 | 0.82 | 0.76 | 0.605 | 0.017 |

db2-WANFIS-25 | 0.023 | 0.94 | 0.85 | 0.892 | 2.1 × 10^{−10} | 0.068 | 0.84 | 0.77 | 0.666 | 0.051 |

coif2-WANFIS-43 | 0.029 | 0.91 | 0.83 | 0.821 | 3 × 10^{−11} | 0.060 | 0.87 | 0.80 | 0.745 | 0.015 |

**Table 6.**Uncertainty analysis of the best-selected models during the testing period for daily SSC predictions.

Model | W_{e} (g/L) | CI_{e}(g/L) |
---|---|---|

ANN-3 | ±0.0096 | −0.0125 to 0.0067 (0.0192) |

Haar-WANN-121 | ±0.0084 | −0.0071 to 0.0097 (0.0168) |

db2-WANN-21 | ±0.0074 | −0.0100 to 0.0049 (0.0149) |

coif2-WANN-11 | ±0.0080 | −0.0118 to 0.0042 (0.0160) |

ANFIS-29 | ±0.0096 | 0.0226 to 0.0034 (0.0260) |

Haar-WANFIS-27 | ±0.0091 | −0.0054 to 0.0128 (0.0182) |

db2-WANFIS-25 | ±0.0083 | 0.0028 to 0.0193 (0.0221) |

coif2-WANFIS-43 | ±0.0073 | −0.0041 to 0.0105 (0.0146) |

Inputs | Training | Testing | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

RMSE (g/L) | r | WI | CE | PARE (%) | RMSE (g/L) | r | WI | CE | PARE (%) | |

R_{t}, R_{t − 1,} R_{t − 2}, R_{t − 3}, Q_{t,} S_{t − 1} | 0.040 | 0.81 | 0.78 | 0.655 | 0.0017 | 0.078 | 0.75 | 0.73 | 0.560 | −0.013 |

R_{t − 1,} R_{t − 2}, R_{t − 3}, Q_{t,} S_{t − 1} | 0.043 | 0.78 | 0.72 | 0.612 | 0.0037 | 0.091 | 0.71 | 0.67 | 0.409 | 0.032 |

R_{t}, R_{t − 2}, R_{t − 3}, Q_{t,} S_{t − 1} | 0.040 | 0.82 | 0.77 | 0.666 | 0.0015 | 0.080 | 0.73 | 0.72 | 0.539 | 0.008 |

R_{t}, R_{t − 1,} R_{t − 3}, Q_{t,} S_{t − 1} | 0.043 | 0.79 | 0.68 | 0.613 | 0.0138 | 0.089 | 0.66 | 0.68 | 0.427 | −0.054 |

R_{t}, R_{t − 1,} R_{t − 2}, Q_{t,} S_{t − 1} | 0.045 | 0.77 | 0.63 | 0.570 | 0.0327 | 0.096 | 0.68 | 0.67 | 0.342 | 0.054 |

R_{t}, R_{t − 1,} R_{t − 2}, R_{t − 3}, S_{t − 1} | 0.041 | 0.80 | 0.73 | 0.639 | 0.0207 | 0.087 | 0.69 | 0.69 | 0.455 | −0.046 |

R_{t}, R_{t − 1,} R_{t − 2}, R_{t − 3}, Q_{t} | 0.045 | 0.75 | 0.69 | 0.563 | 0.0071 | 0.108 | 0.45 | 0.59 | 0.160 | −0.027 |

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## Share and Cite

**MDPI and ACS Style**

Bajirao, T.S.; Kumar, P.; Kumar, M.; Elbeltagi, A.; Kuriqi, A.
Superiority of Hybrid Soft Computing Models in Daily Suspended Sediment Estimation in Highly Dynamic Rivers. *Sustainability* **2021**, *13*, 542.
https://doi.org/10.3390/su13020542

**AMA Style**

Bajirao TS, Kumar P, Kumar M, Elbeltagi A, Kuriqi A.
Superiority of Hybrid Soft Computing Models in Daily Suspended Sediment Estimation in Highly Dynamic Rivers. *Sustainability*. 2021; 13(2):542.
https://doi.org/10.3390/su13020542

**Chicago/Turabian Style**

Bajirao, Tarate Suryakant, Pravendra Kumar, Manish Kumar, Ahmed Elbeltagi, and Alban Kuriqi.
2021. "Superiority of Hybrid Soft Computing Models in Daily Suspended Sediment Estimation in Highly Dynamic Rivers" *Sustainability* 13, no. 2: 542.
https://doi.org/10.3390/su13020542