# A Novel Comprehensive Evaluation Method for Estimating the Bank Profile Shape and Dimensions of Stable Channels Using the Maximum Entropy Principle

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## Abstract

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_{50})) related to the intended problem. An explicit and simple equation of the $\overline{{S}_{t}}$ of banks and the geometric and hydraulic parameters of flow is introduced based on the GEP in combination with the previous shape profile equation related to previous researchers. Therefore, a reliable numerical hybrid model is designed, namely Entropy-based Design Model of Threshold Channels (EDMTC) based on entropy theory combined with the evolutionary algorithm of the GEP model, for estimating the bank profile shape and also dimensions of threshold channels. A wide range of laboratory and field data are utilized to verify the proposed EDMTC. The results demonstrate that the used Shannon entropy model is accurate with a lower average value of Mean Absolute Relative Error (MARE) equal to 0.317 than a previous model proposed by Cao and Knight (1997) (MARE = 0.98) in estimating the bank profile shape of threshold channels based on entropy for the first time. Furthermore, the EDMTC proposed in this paper has acceptable accuracy in predicting the shape profile and consequently, the dimensions of threshold channel banks with a wide range of laboratory and field data when only the channel hydraulic characteristics (e.g., Q and d

_{50}) are known. Thus, EDMTC can be used in threshold channel design and implementation applications in cases when the channel characteristics are unknown. Furthermore, the uncertainty analysis of the EDMTC supports the model’s high reliability with a Width of Uncertainty Bound (WUB) of ±0.03 and standard deviation (${S}_{d}$) of 0.24.

## 1. Introduction

_{c}) to the corresponding lateral distance of this depth from the central channel axis (L). Therefore, if the channel dimension values are not specified, the $\overline{{S}_{t}}$ value cannot be obtained. Therefore, a novel relationship would have existed to estimated $\overline{{S}_{t}}$ values based on available datasets (not only channel dimensions).

_{50}) are available). This relationship is combined with Vigilar and Diplas’ [11] polynomial equation to present an equation for estimating the stable free surface width based on the relationship between δ and K. The EDMTC proposed in this paper is used together with the bank profile shape equation to obtain the channel bank dimensions.

## 2. Literature Review

_{50}. Diplas [21] used an analytical model with their experimental data and proposed a special case of Ikeda’s [20] equation as an exponential function for a bank profile shape. Pizzuto [22] examined the stability criterion using an analytical solution of the widening process at the free water surface. Pizzuto [22] considered the shear stress redistribution due to lateral diffusion and reported an exponential function for a bank profile after channel widening stops. Diplas and Vigilar [23] presented a numerical model to assess the difference between the shape of threshold channels and a previous conventional shape (cosine) for banks. They stated that with particles that do not move along the banks, the transverse slope of the banks should be milder, in which case a wider and deeper channel would form. Hence, they introduced a fifth-degree polynomial profile shape of stable channel banks. Vigilar and Diplas [11,24,25] provided graphs for use to predict the dimensions and profile shapes of stable channel banks with a third-degree polynomial equation. This equation can accurately predict the bank profile shape, because it is in accordance with the results obtained with the equations of several other researchers who have used various other methods [26,27]. Babaeyan [7] did an extensive laboratory study and according to their observational data introduced a hyperbolic bank profile shape. Cao and Knight [28] were the first to examine the shape of bank profiles using the entropy concept. By applying the shape equation obtained with the maximum entropy principle, they reported a parabolic equation. In solving their entropy equation, the Lagrange multiplier (λ) contained within were tested numerically. The equation was validated according to Chow’s [29] definition of natural rivers considering a value of zero for λ. Cao and Knight [28] emphasized the need to further consider the physical concept of multiplier λ. Following Cao and Knight’s [28] brief study, no other study has been based on the entropy concept to predict the ${S}_{t}$ and hence the bank profile shape of stable channels. Gholami et al. [30,31,32,33,34] assessed the ability of different artificial intelligence (AI) methods in the estimation of bank profile shapes of threshold channels. They referred to high efficiency in these methods in estimation and the necessity of further researches about on forming stable shape of bank profiles.

## 3. Materials and Methods

#### 3.1. Maximum Entropy Principle in Estimating the Transverse Slope of Stable Banks

_{c}), which is equal to μ (the submerged static coefficient of Coulomb friction).

_{c}is the dimensionless lateral distance from the channel centerline and y* = y/h

_{c}is the dimensionless vertical boundary level. The Lagrange multiplier is a key component of the maximum entropy principle. In the following, Gholami et al. [54] presented an equation based on the maximum entropy principle to caculate λ numerically [54] which is explained in summary in the following. Accordingly, by using the Lagrange Multiplier Method (LMM) and variable calculation technique [39,60,61], the equation below is obtained for $p({S}_{t})$:

_{1}is Lagrange multipliuer and equal to: λ

_{1}= ln[λ/(e

^{λμ}− 1)] + 1.

_{c}, L, and ${S}_{t}{}^{+}$ (=μ) are known, the $\overline{{S}_{t}}$ value along the banks is obtained by assuming the uniform distribution of ${S}_{t}$ as equal to the h

_{c}/L ratio. Therefore, λ is obtained by numerically solving Equation (8). Then, the ${S}_{t}$ distribution of stable banks can be computed according to Equation (2). Moreover, physical justifications of λ multiplier and the effect of different hydraulic and geometric parameters on it is investigated in Gholami et al. [10]. On the other hand, the ${S}_{t}$ at each point on the channel banks is formulated as ${S}_{t}=dy/dx$, where y is the vertical boundary level of the points. By integrating this, the bank profile shape equation for threshold channels becomes Equation (10), where the integral constant (C) is obtained by applying the boundary condition at the channel centerline (x and y = 0).

_{c}) are not specified, it is not possible to estimate λ and hence, the ${S}_{t}$ and y values. Therefore, in this paper, the next section presents a numerical model for when the channel dimensions are not specified and only Q and d

_{50}are known from the problem condition.

#### 3.2. Calculating μ

_{50}) can be utilized in the current study to compute φ in uniform sediments [10,27,54]:

_{50}should be inserted in centimeters.

#### 3.3. Entropy-Based Design Model of Threshold Channels (EDMTC)

_{c}/L when the values of h

_{c}, and L (=B/2) are known. Accordingly, if the h

_{c}and B values are not known, it is not possible to calculate $\overline{{S}_{t}}$. In this section, an explicit relationship will be provided to calculate the $\overline{{S}_{t}}$ value for the cases that the channel dimensions (h

_{c}, B) are not available.

_{50}and μ values are determined and a relationship for the $\overline{{S}_{t}}$ value based on these parameters is applied to calculate the $\overline{{S}_{t}}$ value for any other data where the channel dimensions are not specified. Accordingly, considering Q, d

_{50}and μ parameters as input parameters and $\overline{{S}_{t}}$ as output parameter based on a numerical GEP model (Figure 2) [32,63,64] provide a relationship for predicting $\overline{{S}_{t}}$ in the form of Equation (12):

_{50}and μ (=${S}_{t}{}^{+}$) the value of $\overline{{S}_{t}}$ is calculated using Equation (12). Now by knowing the $\overline{{S}_{t}}$ value for any channel whose stability dimensions are not specified, in addition to bank profile shape, the width and depth of the channel after stability can be determined. To do this, $\overline{{S}_{t}}$ can be calculated by using the equations presented by former researchers who have applied analytical and theoretical frameworks to derive the relationships. As stated, the polynomial shape proposed by some researchers is an acceptable shape than the previous classic cosine, parabolic, and exponential forms [23]. Therefore, in the present study, the polynomial function provided by Vigilar and Diplas [11] is used to estimate the bank profile shape of stable channels as follows [11]:

_{0}, a

_{1}, a

_{2}and a

_{3}depend on the values of δ*

_{cr}and μ, which are obtained from Table 1 for each given dataset [11]. δ*

_{cr}is the dimensionless critical stress depth (δ*

_{cr}= δ

_{cr}/h

_{c}) in critical condition of sediments in the bank profile. In this case, the shear stress depth (δ′) is δ′ = τ/ρgS, where τ is the shear stress along the channel and S is the longitudinal slope of the water surface. The value of δ*

_{cr}can be obtained according to the (μ − δ*

_{cr}) figures related to Vigilar and Diplas [11].

_{50}and μ (=${S}_{t}{}^{+}$) the value of $\overline{{S}_{t}}$ is calculated using Equation (12). Then, Equation (15c) is used to obtain the value of B* based on obtained $\overline{{S}_{t}}$ values according to Equation (12). Accordingly, in this study, the EDMTC (Figure 2) is presented to predict the dimensions and shape of bank profiles using the entropy principle. The value of x* (lateral distance from the channel axis) is selected for a specific range of arbitrary x* values at a distance of $0\le x{*}_{i}\le 0.5B*(=L)$. The values of y* obtained by the entropy facilitate plotting the bank shape profiles against different x

_{i}. Figure 2 shows the flowchart of the GEP model and model developed in the present study (EDMTC) to predict the shape and dimensions of threshold channels.

#### 3.4. Experimental Data

_{50}values in the channel as well as geometric conditions of the laboratory flumes used with each data series. Furthermore, several tests were carried out for different discharge rates with each data series, and the channels had different conditions until reaching equilibrium state. In each observational data series, in addition to the channel dimensions (B and h

_{c}) the coordinate data of the points in stable bank profiles (x, y) were extracted for some discharge values as well. Moreover, all experiments were done in laboratory flumes with different aspect ratios (B/h

_{c}= α) in the range (4–30). In each test, the sediment sizes selected were somewhat course, so the corresponding proportional discharge in the channels would cause no movement of sediment particles in the channels. Hence, the stresses on the walls and channel bed were respectively less and more than the critical stress until threshold channel conditions would govern. Table 2 summarizes the hydraulic and geometric conditions for the data used.

#### 3.5. Used Data in Modeling

_{50}values in the channel, as well as the geometric conditions of the laboratory flumes used in each data series, are different. Furthermore, in each seven available observational data series (Mikhailova et al. 1980; Ikeda 1981; Diplas 1990; Babaeyan 1996; Macky 1999; Hassanzadeh et al. 2014; and Khodashenas 2016), there are several runs related to them according below with different discharges, therefore, the stable channel shape formed on banks in each observed run is different.

- Ikeda (1981) → one run as
**S3**(8 samples) - Diplas (1990) → one run as
**S4**(25 samples) - Babaeyan (1996) → one run as
**S5**(8 samples) - Macky (1999) → one run as
**S6**(101 samples) - Hassanzadeh et al. (2014) → two runs as
**S7**(33 samples) and**S8**(38 samples) - and Khodashenas (2016) → four runs as
**S9**(44 samples),**S10**(33 samples),**S11**(57 samples) and**S12**(20 samples)

#### 3.6. Evaluation of Model Efficiency

^{2}), Root Mean Squared Error (RMSE), Mean Absolute Relative Error (MARE), Mean Absolute Error (MAE), and Bias. These evaluation criteria are defined by Equations (16)–(20):

_{i}and x

_{i}denote the estimated and observed values, $\overline{y}$ represents the mean modeled values and n is the sample size. The closer the R

^{2}coefficient is to the unit value (1), the higher the agreement there is between the observed and predicted values. The closer the results of MARE, RMSE, Bias, and MAE indices are to zero, the higher the estimation accuracy is as well. Positive and negative Bias values imply model over and underestimation, respectively [69,70,71]. Therefore, computing several evaluation criteria can better reveal the model performance [72,73].

## 4. Results

#### 4.1. Entropy Model in Predicting Bank Profile Shapes

_{c}/L. Using obtained λ value, the y value is computed based on entropy method by solving Equation (10). The y* distribution obtained by Equation (10) corresponding each x* value is drawn for each data series in Figure 3. Moreover, the results of Cao and Knight’s [28] model (CKM) (according to Equation (5)) are extracted and their proposed bank profile shape is drawn in Figure 3 to evaluate the entropy model performance. Table 3 contains the different error indices for entropy model and CKM. Figure 3 indicates that entropy model exhibits acceptable conformity with the corresponding observational data series in predicting the vertical boundary level and hence, estimates the bank profile shape with low error values. According to all data series, entropy model is able to estimate the governing bank profile shape trend with lower MARE and RMSE values equal to 0.317 and 0.08 better than CKM with 0.981 and 0.363 values respectively. Figure 3 also shows that for two data series, i.e., S1 and S2 (Mikhailova et al.’s [65] data), CKM has high error values in y* estimation and high accuracy in the area near the free water surface, where high MARE values in the 2–4 range are observed for these data series. However, the proposed entropy model is able to detect the bank profile shape trend with lower error values (MARE = 0.2 and 0.8 for S1 and S2 datasets respectively) than CKM with 1.95 and 3.95 MARE values, which represents the significant superiority of entropy model. This process is repeated for the S2 and S3 data series. Although CKM exhibits acceptable performance, entropy model is more accurate with lower error values and coincides closely with the observed values (especially in the area near the surface). For the S6 field data series, although both models do not perform well (with close Bias values of −0.31 and 0.45 for entropy model and CKM respectively), entropy model again performs with lower error (MARE = 0.58) than CKM (MARE = 1.03). Furthermore, the high MARE index value for CKM is representative of its inability to estimate low y* values (in the vicinity of channel bed), a problem that is solved by entropy model significantly. Furthermore, the RMSE values of CKM and entropy model which is equal to 0.5 and 0.38 respectively approved the inefficiency of CKM in estimating low y* levels. With data series S7, the improvement of entropy model over CKM by about 60% and 85% in the MARE and RMSE values respectively is observed clearly in Figure 3, as entropy model highly conforms to the observational data with R

^{2}values of 0.98. With Khodashenas’ [68] data (S9–S12), the higher efficiency of entropy model over CKM is evident with lower MARE and RMSE values in entropy model than CKM. Furthermore, entropy model is able to estimate the water surface widening with high y* values well with low values of RMSE and Bias values close to 0. The negative and positive Bias value represents the underestimation and overestimation of the models respectively. As it can be seen in the Bias values, the CKM in most of the datasets have positive Bias values and overestimates the y* values in comparison with the corresponding observed values. It can thus be said that the entropy model proposed in the present study based on the maximum entropy principle is more accurate in the estimating the bank profile shape of stable channels than CKM, which suggests a parabolic curve (Equation (5)) for channel banks. A notable point in this paper is the significant physical effect of λ values on the accurate estimation of the intended variables, which is negligible with CKM. The λ values obtained by entropy model in this study are gathered in Table 3, where it can be seen that this multiplier is in a specified range of −2 to 2 with almost all data series (except with 1–2 data series). Furthermore, the λ values are the same for different runs of one experiment.

#### 4.2. Presenting the Entropy-Based Design Model of Threshold Channels (EDMTC)

#### Evaluation of EDMTC Performance

^{2}index value in this figure is higher than 0.95 for all observational data series, indicating the high EDMTC prediction accuracy. The value of this index is very close to 1 for some of the observational data [20,21,68], signifying very high model conformity to the corresponding observational values. Furthermore, according to the diagrams on the right side of Figure 4, the EDMTC is able to accurately estimate the bank profile shape trend for all data series. Although some differences between the values y* predicted by the model and the observational values are seen, it is notable that EDMTC is able to model the vertical bank elevation (from the channel center on the bed to the free water surface margins) and the water surface widening near the water surface levels similar to the corresponding observational values. The error index values in Table 4 are also validated accordingly. This table shows that the MARE values for all datasets are 0.3–0.5, which is close to 0. This index indicates the accuracy of the proposed EDMTC in predicting the vertical elevation of banks as well as the free water surface width in stable channels. An important point is that the proposed EDMTC predicts the profile shape trend successfully and can therefore be used to design the width and depth (dimensions) of stable channels when only flow inputs such as Q, d

_{50}and μ are known. The high accuracy of this model is confirmed, and achieving such a model with the least parameters to predict the dimensions and cross-sectional bank shapes formed in stable channels is of considerable importance. Also, EDMTC not only considers the geometric conditions of the channel cross sections but also involves the hydraulic conditions of the problem (by using Vigilar and Diplas’ [11] equation), which is one of the notable features of this model. Based on most observational data series, the estimated channel width is very similar to the observational values (in some cases it is slightly less). For example, for the EDMTC profile predictions based on the observational data from Diplas [21], Babaeyan [7], and Hassanzadeh et al. [67], the water surface width is estimated very close to the observed values. Furthermore, for most observational datasets, the proposed model estimates greater values for the vertical elevation of the water surface, although the estimated profile trend fits the observational values perfectly. The partial error values of EDMTC that are mostly seen in the areas near the channel bed and the free water surface with some of the datasets can be considered measurement errors of the observational data [74]. For some data, e.g., Hassanzadeh et al. [67] and Khodashenas [68] this error is seen at the channel bed. Additionally, Figure 4 shows that EDMTC based on Khodashenas’ [68] data estimates lower y* than the actual values, which results in a negative Bias and an absolute error increase of 14% in MAE value according Table 4 (MAE represents the absolute magnitude of the difference between observational values and the model). It is worth noting that the EDMTC can estimate a more logical shape than the profile derived from the corresponding observational values, which has a uniform distribution from the bed to the water surface. With the rest of the data series, EDMTC estimates roughly higher partial values equal to the observational values for y*, as the RMSE error value is about 0.9–0.13, which is acceptable. Therefore, EDMTC with low average error values (MARE = 0.55 and MAE = 0.19) is generally highly accurate in predicting bank profiles and stable channel dimensions.

#### 4.3. Uncertainty Analysis of the Proposed EDMTC and GEP Model

## 5. Conclusions

_{c}) was designed based on the maximum entropy principle in combination with the GEP regression model for cases when only the Q and d

_{50}are known as problem conditions. The results indicate that the entropy model is capable of predicting the bank profile shape trend with acceptable error values (MARE = 0.317, RMSE = 0.09) according to the experimental data in comparison with the Cao and Knight’s [28] model (MARE = 0.317, RMSE = 0.09). Therefore, the λ multiplier has a significant role in determining the transverse slope and consequently the vertical elevation of banks, and the physical meaning of λ is associated with the hydraulic parameters governing the problem. The EDMTC proposed in this study with R

^{2}greater than 0.95 and MAE in the 0.076–0.436 range for different observational data series is able to predict the bank profile shape trend as well as the free water surface level in threshold channels. In addition, the uncertainty analysis of EDMTC demonstrated that more than 95% of predicted and observed data are within the CB with low WUB, and the model reliability is largely assured. The EDMTC computational model presented in this paper can be used widely to predict stable channel profiles when the given problem information only includes the Q and d

_{50}. This study was developed on Shannon entropy concept, it is suggested to improve the obtained results with other generalized entropies. It is further recommended that other equations provided by different researchers be used to estimate the free surface width of channels. Regression and AI models based on more field data also ought to be used to estimate the mean transverse slope of banks as well as other entropy model types to examine the accuracy of the model presented in this study.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Flowchart of the proposed Entropy-based Design Model of Threshold Channels (EDMTC) computational procedure for designing the dimensions and shape of threshold channels in the present study.

**Figure 3.**Bank profile shape predicted by developed entropy model and Cao and Knight’s [28] model (CKM) for different observational data series (S1–S12).

**Figure 4.**Comparison of values predicted for the vertical boundary level of stable channels by the EDMTC proposed in the present study using scatter plots (left side) and cross-sectional profile shapes (right side) for different observational data: (

**a**) Ikeda [20]-S3, (

**b**) Babaeyan [7]-S5, (

**c**) Diplas [21]-S4, (

**d**) Hassanzadeh et al. [67]-S7, (

**e**) Khodashenas [68]-S9, and (

**f**) Khodashenas [68]-S12.

**Figure 5.**CB (95%) ranges for the observational values and values predicted by EDMTC for the vertical boundary elevation of stable channels based on different datasets of (

**a**) Ikeda [20] (S3), (

**b**) Diplas [21] (S4), (

**c**) Babaeyan [7]-S5, (

**d**) Khodashenas [68]-S9, (

**e**) Khodashenas [68]-S10, and (

**f**) Khodashenas [68]-S11.

a_{0} | a_{1} | a_{2} | a_{3} | δ*_{cr} |
---|---|---|---|---|

μ = 0.4 | ||||

1.0001 | −0.0135 | −0.0411 | 0 | 0.93 |

1.0004 | −0.0236 | −0.0412 | 0 | 0.935 |

1.0008 | −0.0307 | −0.0412 | 0 | 0.94 |

1.0009 | −0.0342 | −0.0413 | 0 | 0.945 |

μ = 0.55 | ||||

1.0003 | −0.018 | −0.0503 | −0.0029 | 0.9 |

1.0006 | −0.0299 | −0.0527 | −0.0027 | 0.905 |

1.0008 | −0.0366 | −0.0547 | −0.0025 | 0.91 |

1.001 | −0.0416 | −0.0565 | −0.0022 | 0.915 |

1.0011 | −0.0463 | −0.0586 | −0.0019 | 0.921 |

μ= 0.65 | ||||

1.0006 | −0.0278 | −0.0543 | −0.006 | 0.885 |

1.001 | −0.0444 | −0.06 | −0.0054 | 0.895 |

1.0013 | −0.0529 | −0.0647 | −0.0048 | 0.905 |

1.0041 | −0.0556 | −0.0665 | −0.0045 | 0.909 |

μ= 0.76 | ||||

1.0009 | −0.0365 | −0.0544 | −0.0105 | 0.87 |

1.0014 | −0.0531 | -0.061 | −0.0101 | 0.88 |

1.0017 | −0.0621 | −0.0662 | −0.0095 | 0.89 |

1.0018 | −0.0662 | −0.0701 | −0.009 | 0.897 |

μ= 0.84 | ||||

1.0011 | −0.0418 | −0.0516 | −0.0146 | 0.86 |

1.0016 | −0.0594 | −0.059 | −0.0143 | 0.87 |

1.002 | −0.0697 | −0.0634 | −0.0141 | 0.88 |

1.0021 | −0.0742 | −0.0708 | −0.013 | 0.89 |

μ= 1.0 | ||||

1.0016 | −0.0571 | −0.0466 | −0.0233 | 0.845 |

1.0022 | −0.0738 | −0.0531 | −0.0237 | 0.855 |

1.0025 | −0.0828 | −0.0589 | −0.0236 | 0.865 |

1.0028 | −0.0884 | −0.0656 | −0.023 | 0.875 |

1.0028 | −0.0892 | −0.0683 | −0.0226 | 0.878 |

Researchers | Runs. No. | No. of Series | d_{50} [mm] | Discharge (Q) [L/s] | Water Surface Half-Width (B/2) [cm] | Central Water Depth (h_{c}) [cm] |
---|---|---|---|---|---|---|

Mikhailova et al. [65] | 2 | S1 | 0.2 | 65 | 112 | 10.4 |

S2 | 0.2 | 69 | 132.5 | 14.4 | ||

Ikeda [20] | 1 | S3 | 1.3 | 16.28 | 24.8 | 3.54 |

Diplas [21] | 1 | S4 | 1.9 | 12.526 | 33 | 3.85 |

Babaeyan [7] | 1 | S5 | 1 | 2.5 | 52.6 | 2.63 |

Macky [66] (Field data) | 1 | S6 | 3.42 | 64.3 | 127 | 3.7 |

Hassanzadeh et al. [67] | 2 | S7 | 1.2 | 11.09 | 32 | 8.6 |

S8 | 1.6 | 20.07 | 40.6 | 10.9 | ||

Khodashenas [68] | 4 | S9 | 0.53 | 6.2 | 21.7 | 8 |

S10 | 0.53 | 2.57 | 16 | 6.3 | ||

S11 | 0.53 | 2.18 | 17 | 6.12 | ||

S12 | 0.53 | 1.157 | 9.5 | 3.7 |

**Table 3.**Assessment of the efficiency of developed entropy model (DEM) and CKM compared with different observational data series according to different error indices and λ values related to DEM in this paper.

MARE | RMSE | Bias | R^{2} | λ | |||||
---|---|---|---|---|---|---|---|---|---|

Data Series | DEM | CKM | DEM | CKM | DEM | CKM | DEM | CKM | DEM |

S1 | 0.254 | 1.95 | 0.103 | 1.31 | −0.04 | 0.99 | 0.93 | 0.981 | −5.56 |

S2 | 0.86 | 3.95 | 0.057 | 0.7 | −0.036 | 0.47 | 0.98 | 0.988 | −4.26 |

S3 | 0.228 | 0.47 | 0.037 | 0.141 | 0.022 | 0.116 | 0.99 | 0.981 | −1.62 |

S4 | 0.15 | 0.11 | 0.053 | 0.08 | −0.05 | 0.064 | 0.99 | 0.997 | −1.75 |

S5 | 0.43 | 0.42 | 0.1 | 0.135 | −0.08 | 0.114 | 0.99 | 0.988 | 2.11 |

S6 | 0.58 | 1.03 | 0.38 | 0.5 | −0.31 | 0.45 | 0.96 | 0.957 | 1.5 |

S7 | 0.147 | 0.86 | 0.056 | 0.37 | 0.045 | 0.35 | 0.98 | 0.966 | −2.46 |

S8 | 0.315 | 0.99 | 0.109 | 0.34 | 0.098 | 0.32 | 0.97 | 0.95 | −2.2 |

S9 | 0.26 | 0.50 | 0.044 | 0.184 | 0.008 | −0.148 | 0.98 | 0.989 | 1.72 |

S10 | 0.18 | 0.56 | 0.028 | 0.24 | −0.01 | −0.192 | 0.99 | 0.987 | 2.2 |

S11 | 0.23 | 0.46 | 0.05 | 0.14 | 0.03 | −0.108 | 0.99 | 0.996 | 1.4 |

S12 | 0.17 | 0.47 | 0.05 | 0.22 | 0.03 | −0.16 | 0.985 | 0.996 | 2.4 |

Averaged | 0.317 | 0.981 | 0.08 | 0.363 | −0.02 | 0.189 | 0.978 | 0.981 | - |

**Table 4.**Evaluation of the EDMTC proposed in the present study in estimating the dimensions of stable channels in comparison with several available observational data series.

Dataset | R^{2} | MARE | RMSE | MAE | Bias |
---|---|---|---|---|---|

Ikeda [20] (S3) | 0.995 | 0.357 | 0.098 | 0.078 | 0.064 |

Diplas [21] (S4) | 0.991 | 0.186 | 0.132 | 0.097 | 0.094 |

Babaeyan [7] (One set) (S5) | 0.961 | 0.400 | 0.124 | 0.095 | −0.095 |

Macky [66] (S6) | 0.942 | 0.568 | 0.556 | 0.381 | 0.380 |

Hassanzadeh et al. [67] (S7) | 0.986 | 1.164 | 0.456 | 0.436 | 0.436 |

Hassanzadeh et al. [67] (S8) | 0.981 | 1.146 | 0.380 | 0.364 | 0.364 |

Khodashenas [68] (S9) | 0.992 | 0.426 | 0.127 | 0.109 | −0.109 |

Khodashenas [68] (S10) | 0.979 | 0.473 | 0.169 | 0.143 | −0.143 |

Khodashenas [68] (S11) | 0.994 | 0.361 | 0.096 | 0.076 | −0.076 |

Khodashenas [68] (S12) | 0.995 | 0.475 | 0.193 | 0.147 | −0.147 |

Average | 0.9816 | 0.5556 | 0.2331 | 0.1926 | 0.0768 |

**Table 5.**Uncertainty analysis for the Gene Expression Programming (GEP) model in $\overline{{S}_{t}}$ prediction according to Equation (12) and EDMTC.

Model | Datasets | Sample Number | ${\mathit{S}}_{\mathit{d}}$ | MPE | WUB | ${\overline{\mathit{d}}}_{\mathit{x}}$ | CB |
---|---|---|---|---|---|---|---|

EDMTC | Ikeda [20] (S3) | 8 | 0.08 | −0.064 | ±0.07 | 0.065 | −0.13 to 0.00 |

Diplas [21] (S4) | 25 | 0.09 | −0.094 | ±0.04 | 0.09 | −0.13 to −0.05 | |

Babaeyan [7] (S5) | 8 | 0.08 | 0.095 | ±0.075 | 0.095 | +0.02 to +0.17 | |

Khodashenas [68] (S9) | 44 | 0.07 | 0.109 | ±0.02 | 0.11 | +0.09 to +0.13 | |

Khodashenas [68] (S10) | 33 | 0.09 | 0.143 | ±0.035 | 0.145 | +0.11 to +0.18 | |

Khodashenas [68] (S12) | 20 | 0.13 | 0.147 | ±0.06 | 0.15 | +0.09 to +0.21 | |

All datasets | 266 | 0.33 | −0.14 | ±0.04 | 0.14 | −0.18 to −0.10 | |

GEP, Equation (12) | All datasets | 20 | 0.02 | -0.009 | ±0.01 | ±0.01 | −0.02 to 0.00 |

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**MDPI and ACS Style**

Bonakdari, H.; Gholami, A.; Mosavi, A.; Kazemian-Kale-Kale, A.; Ebtehaj, I.; Azimi, A.H.
A Novel Comprehensive Evaluation Method for Estimating the Bank Profile Shape and Dimensions of Stable Channels Using the Maximum Entropy Principle. *Entropy* **2020**, *22*, 1218.
https://doi.org/10.3390/e22111218

**AMA Style**

Bonakdari H, Gholami A, Mosavi A, Kazemian-Kale-Kale A, Ebtehaj I, Azimi AH.
A Novel Comprehensive Evaluation Method for Estimating the Bank Profile Shape and Dimensions of Stable Channels Using the Maximum Entropy Principle. *Entropy*. 2020; 22(11):1218.
https://doi.org/10.3390/e22111218

**Chicago/Turabian Style**

Bonakdari, Hossein, Azadeh Gholami, Amir Mosavi, Amin Kazemian-Kale-Kale, Isa Ebtehaj, and Amir Hossein Azimi.
2020. "A Novel Comprehensive Evaluation Method for Estimating the Bank Profile Shape and Dimensions of Stable Channels Using the Maximum Entropy Principle" *Entropy* 22, no. 11: 1218.
https://doi.org/10.3390/e22111218