# Uncertainty Assessment of WinSRFR Furrow Irrigation Simulation Model Using the GLUE Framework under Variability in Geometry Cross Section, Infiltration, and Roughness Parameters

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## Abstract

**:**

^{2}) were used in GLUE analysis for selecting behavioral estimations of the model outputs. The uncertainty of the WinSRFR model was investigated under two furrow irrigation system conditions, closed end and open end. The results showed the likelihood measures considerably influence the width of uncertainty bounds. WinSRFR outputs have high uncertainty for cross section parameters relative to soil infiltration and roughness parameters. Parameters of soil infiltration and roughness coefficient play an important role in reducing the uncertainty bound width and number of observations, especially by filtering non-behavioral data using likelihood measures. The simulation errors of advance front curve and runoff hydrograph outputs with a PBIAS function were relatively lower and stable compared with other those of the likelihood functions. The 95% prediction uncertainties (95PPU) of the advance front curve were calculated to be 87.5% in both close-ended and open-ended conditions whereas, it was 91.18% for the runoff hydrograph in the open-ended condition. Additionally, the NSE likelihood function more explicitly determined the uncertainty related to flow depth hydrograph estimations. The outputs of the model showed more uncertainty and instability in response to variability in soil infiltration parameters than the roughness coefficient did. Therefore, applying accurate field methods and equipment and proper measurements of soil infiltration is recommended. The results highlight the importance of accurately monitoring and determining model input parameters to access a suitable level of WinSRFR uncertainty. In conclusion, considering and analyzing the uncertainty of input parameters of WinSRFR models is critical and could provide a reference to obtain realistic and stable furrow irrigation simulations.

## 1. Introduction

- Evaluating the outputs uncertainty of the WinSRFR model in an innovative way, including advance curve, flow depth hydrograph, and runoff hydrograph, using the GLUE framework in response to Kostiakov–Lewis infiltration equation, Manning’s roughness coefficient, and cross-sectional parameters sets based on the experimental data of a furrow irrigation system.
- Assessing the reliability predictions of WinSRFR model for two open- and closed-ended of furrow irrigating systems.
- Investigating the effects of three likelihood functions, the coefficient of determination (R
^{2}), Nash–Sutcliffe efficiency (NSE), and percentage bias (PBIAS), on the parameter set performance and uncertainty analysis of the WinSRFR model.

## 2. Materials and Methods

#### 2.1. Description of WinSRFR Model

^{3}·s

^{−1}), $t$ is the irrigation time (s), $v$ is the flow velocity (m·s

^{−1}), $x$ is distance from the beginning of the furrow (m), ${I}_{x}$ is the infiltration rate (m

^{3}·s

^{−1}·m

^{−1}), $g$ is the acceleration of gravity (m·s

^{−2}), ${S}_{0}$ is the bed slope (m·m

^{−1}), ${S}_{f}$ is the energy line slope (m·m

^{−1}), and $A$ is the flow cross-section (m

^{2}). Saint-Vanant equations do not have analytical solutions. Therefore, different software packages have been presented considering simplified hypotheses and numerical methods to solve these equations simultaneously and simulate surface irrigation. WinSRFR software 5.1, which was developed in 2004, is a one-dimensional mathematical model used for analyzing and simulating surface irrigation. This software has many applications in the design and management of surface irrigation and consists of four parts for the hydraulic analysis of surface irrigation, including simulation, event analysis (field evaluation), physical design, and operation analysis [8]. WinSRFR software performs the necessary calculations using zero-inertia model (in which inertia has been omitted and is applicable to farms with closed ends and slopes of less than 0.004), as well as kinematic-wave model (in which flow depth gradient and inertia have been omitted and are applicable to slopes greater than or equal to 0.004). WinSRFR software could estimate the amount of infiltrated water by different equations such as Kostiakov, Kostiakov–Lewis, and Green-Ampt [41]. In the present study, the Kostiakov–Lewis infiltration equation was used. This model could simulate furrow irrigation on negative slopes and adapt to variable inflow rates [42]. However, the simulations obtained by this model may face uncertainties due to a large number of software input parameters, as well as temporal and spatial variations in the parameters such as infiltration, surface roughness, and furrow cross-sectional dimensions.

#### 2.2. Description of Important Input Parameters of WinSRFR Model

#### 2.2.1. Geometry Cross Section

#### 2.2.2. Kostiakov–Lewis Infiltration Function

^{3}·m

^{−1}), t is the time period of infiltration (min), f

_{0}is the steady infiltration rate (m

^{3}·m

^{−1}·min

^{−1}), and k and a are the fitted empirical coefficients. Furthermore, the steady infiltration rate is calculated as follows based on the equation provided in [25].

_{in}is the inflow rate (m

^{3}·min

^{−1}), Q

_{out}is the outflow rate (m

^{3}·min

^{−1}), and L is the furrow length (m).

#### 2.2.3. Manning’s Roughness Coefficient

^{2}), S is the longitudinal slope of the water surface (m·m

^{−1}), Q is the inflow rate (m

^{3}·s

^{−1}), and P is the wetted perimeter of the furrow cross section (m).

#### 2.3. Generalized Likelihood Uncertainty Estimation (GLUE)

^{2}, NSE, and PBIAS were used to assess the performance of each parameter set. The likelihood functions are as follows.

^{2}, NSE, and PBIAS were from 0 to 1, −∞ to 1, and −1 to 1, respectively. In this study, simulations that had a likelihood value of NSE that was less than or equal to 0.5 were rejected as non-behavioral, and the others (NSE > 0.5) were retained for further analyses. Additionally, the threshold values of R

^{2}> 0.6 and −0.25 < PBIAS < 0.25 were used to identify behavioral simulations. The behavioral parameter sets were applied to extract the posterior distributions and calibration parameters of the Kostiakov–Lewis infiltration equation, Manning’s roughness coefficient, and cross section. Step 3: Simulation weights of the behavioral parameter sets were normalized to unity, and the cumulative weighted distributions of estimations were used to evaluate the uncertainty associated with the model simulations.

#### 2.4. Data Sets

#### 2.5. Evaluation Criteria

## 3. Results and Discussion

#### 3.1. Creating Random Sample Sets Using Monte Carlo Simulation

_{0}), and Manning’s roughness coefficient (n). To investigate the effect of these six states on WinSRFR furrow irrigation simulation model estimations, the probability distribution of each parameter was derived (Table 2), and 1000 datasets were produced. The models’ simulations mainly focused on the three plots of the advance front, depth hydrograph, and flow hydrograph. The selection of probability distribution function for each parameter is important to obtain data samples using the corresponding parameter space because it affects the quality of the simulation results [49]. Hence, in the first step of the framework, the probability distribution of geometry cross section, infiltration, and roughness parameters was investigated and applied to provide information on the parameters, and furthermore, the simulations. As seen from Table 2, all BW, SS, a, f

_{0}, and n parameters followed the Wakeby distribution with p-values equal to 0.13, 0.13, 0.12, 0.12, and 0.19, respectively. Additionally, the results of the K-S test illustrated that the Log-Logistic distribution fitted the infiltration parameter of k, which was significant at $\alpha =0.05$. The finding results shows that the probability distributions of related irrigation parameters are free from Normal and Gaussian distributions that str commonly used in order to simplify the procedure [50].

#### 3.2. Simulation Using WinSRFR Furrow Irrigation Model and Uncertainty Analysis

_{0}), open-ended (k, a, f

_{0}), close-ended (n), and open-ended (n) ones covers 100%, 87.5%, 87.5%, and 87.5% of the front curve observation data, respectively. Among the infiltration and roughness parameters, the finding results related to the r-factor value demonstrate the superiority of the model based on the Manning’s parameter, especially for the close-ended condition. The calculated r-factor values for the close-ended (n) one are 44%, 69%, and 15% lower than the close-ended (k, a, f

_{0}), open-ended (k, a, f

_{0}), and open-ended (n) ones, respectively. Although infiltration and roughness models create more accurate results than geometry cross section parameters do in estimating front curve, their use still has high instability and uncertainty. The WinSRFR irrigation model uses empirical relationships between the Kostiakov–Lewis and Manning’s functions for representing infiltration and flow processes. Even if physical-based infiltration models can be applied, their practical applications remain limited. It is partly due to problems in coupling infiltration calculations to surface flow models and the difficulty in considering soil hydraulic properties, geometric properties, and other effective parameters [51]. Similar to the uncertainty results of the front curve, WinSRFR has high instability and uncertainty when it is used to estimate the flow depth hydrograph based on the geometry cross section properties of the furrow (Figure 2). Although the results of the p-factor show that all the observation flow depth data are covered by the 95PPU bound, the uncertainty bounds are wide, indicating the low accuracy of estimations [52]. However, the estimation ranges of flow depth hydrograph for close-ended (SS, BW) and open-ended (SS, BW) ones were wide. These bounds did not contain the measured values and produced large uncertainties.

_{0}) and (n) parameters, most of the observations fell within the confidence intervals of 2.5% and 97.5%. A few observations were higher than those corresponding to the 95PPU limit produced by Manning’s parameter. There is a wide range of estimations in response to the geometry cross section, infiltration, and roughness parameters. This result demonstrates uncertainties in the Kostiakov–Lewis and Manning equations. In addition, the uncertainties may also lead to severe fluctuations in the estimation results, as well as the large variations in the hydrographs patterns in some cases. Therefore, the WinSRFR furrow irrigation model shows some limitations in estimating the advance front curve, flow depth hydrograph, and runoff hydrograph. Cahoon [53] showed that parameters of the Kostiakov–Lewis infiltration function have a strong influence on the advance front curve and flow hydrograph, which is in agreement with the findings of the current study. The results of the current study are in agreement with those stated by Gillies and Smith [54], which reported that soil infiltration parameters of the modified Kostiakov equation are more sensitive to runoff hydrograph than the advance front curve is, indicating the need for the precise measurement of field data. Khoi and Tom [55] presented that the uncertainty of the predicted flow hydrograph using the SWAT model is high, and the model is sensitive to parameters values.

#### 3.3. Effect of Likelihood Measures on Uncertainty

^{2}, and PBIAS are used to investigate their effects on the uncertainty results in WinSRFR furrow irrigation modeling. The percentage of behavioral likelihood, which are derived via GLUE, are given in Table 4. Based on the posterior distribution of geometry cross section parameters, 1000 simulations were made by WinSRFR furrow irrigation modeling. All the simulations were behavioral for estimating the front curve based on the geometry cross section properties in both the open-ended and close-ended conditions (NSE > 0.5, −0.25 < PBIAS < 0.25, R

^{2}> 0.6). The posterior and prior cumulative distributions of the advance front derived using different criteria for T = 134 min at a distance of 168.6 cm in the close-ended furrow irrigation condition are shown in Figure 4. In simulations of furrow irrigation, the difference between the prior and posterior distributions demonstrated that the model was most sensitive to infiltration parameters for front curves. As seen in Figure 4, the difference is greater for the PBIAS criterion. The dotted points in Figure 4 show the distributions of behavioral data for each parameter, whose likelihood measures NSE > 0.5, −0.25 < PBIAS < 0.25, and R

^{2}> 0.6. For the close-ended furrow irrigation condition, the kurtosis in the distribution of the simulation by using roughness parameter and NSE and PBIAS criteria show that this parameter has major effects on advance front curve modeling. Additionally, among the infiltration simulations and dotted plot distribution, the kurtosis of behavioral sets by applying NSE and PBIAS criteria is a more obvious than the other R

^{2}is. Hence, it seems the PBIAS and NSE criteria can provide a more efficient way of modeling the front curve based on roughness and infiltration parameters. The acceptable percentages were 96.61% for the close-ended one (n)-NSE; 86.64% for close-ended one (n)-PBIAS; 56.35% for close-ended one (k, a, f

_{0})-NSE; and 21.54% for close-ended one (k, a, f

_{0})-PBIAS. Similar results were obtained for the open-ended furrow irrigation condition by applying WinSRFR modeling (Figure 5 and Table 4). Based on Figure 6 and Table 4 and considering the kurtosis of behavioral data, using the roughness parameter and NSE criterion can produce a relatively stable flow depth hydrograph estimation for close-ended furrow irrigation condition. The estimations based on the geometry cross section and infiltration properties by applying the NSE criterion were ranked in second place and can be used after roughness-NSE. Additionally, for the open-ended furrow irrigation condition, the kurtosis distribution of behavioral data displays the role of infiltration (NSE and PBIAS) parameters and geometry cross section (NSE) properties on the estimation stability of WinSRFR (Figure 7). The results of behavioral data dispersion on the posterior distribution line of runoff hydrograph show the effect of the Manning equation on WinSRFR estimation, especially after applying the NSE likelihood function (Figure 8).

_{0}), and roughness (n) parameters and affect the model’s output uncertainty. Hence, the question is what criterion is useful to investigate the uncertainty of the WinSRFR model for estimating advance front curves and depth and runoff hydrographs. To answer this question, the Taylor diagram and 95PPU bounds were drawn after applying likelihood functions for removing non-behavioral data. The percentage of covered observation, the width of uncertainty bounds, and statistical characteristics were evaluated.

#### 3.4. Analysis of Accuracy and Uncertainty Associated with Different Parameters

^{2}> 0.6, and −0.25 < PBIAS < 0.25. For this aim, those estimations that satisfied the NSE, PBIAS, and R

^{2}criteria were selected, and the uncertainty and statistical analyses were performed again.

^{2}functions were classified as behavioral data, one Taylor diagram was plotted. The estimations were located between R lines of 0.99 and 1, STD lines of 42.39 and 51.44, and RMSD lines of 1.80 and 6.29 for the advance front curve of close-ended condition (SS, BW). For the estimation of advance front curve using infiltration parameters, the estimations were located between R lines of 0.96 and 1, STD lines of 23.47 and 59.82, and RMSD lines of 0.96 and 25.16 for the NSE function; between R lines of 0.96 and 1, STD lines of 32.49 and 59.82, and RMSD lines of 0.96 and 16.50 for the PBIAS function; between R lines of 0.96 and 1, STD lines of 15.99 and 59.82, and RMSD lines of 0.96 and 32.47 for the R

^{2}function. Additionally, for estimation of the advance front curve using the roughness parameter of Manning equation, the estimations were located between R lines of 0.99 and 1, STD lines of 35.56 and 67.52, and RMSD lines of 1.68 and 19.37 for the NSE function; between R lines of 0.99 and 1, STD lines of 38.77 and 60.74, and RMSD lines of 1.68 and 12.44 for the PBIAS function; between R lines of 0.99 and 1, STD lines of 35.56 and 67.52, and RMSD lines of 1.68 and 19.37 for the R

^{2}function. It is obvious that all estimations in terms of R exhibit high accuracy in predicting advance front curve in the close-ended condition. Taylor diagrams show that (SS, BW), (k, a, f

_{0})-PBIAS, and (n)-PBIAS have high accuracy in recognizing the complex relationships between the input parameters and advance front curve. For a better selection of robust parameters, the values of p-factor and r-factor are given in Table 5. The results show that (k, a, f

_{0})-PBIAS and (n)-PBIAS can cover 87.5% and 87.5% of the observation data, and the widths of the uncertainty bound are 0.51 and 0.40, respectively. By removing non-behavioral data (PBIAS > 0.25 and PBIAS < −0.25), the estimations of (n)-PBIAS reduced the r-factor value from 0.64 to 0.4, while the p-factor was constant (87.5%). The results suggest that if proper soil infiltration parameters are selected and accurate measurements are taken, it is possible to reduce the simulation uncertainty of advance front curves. Therefore, to minimize the simulations uncertainty of the WinSRFR furrow irrigation model, it is important to select appropriate parameters prior to modeling advance front curves. Nie et al. [56] demonstrated that the sensitivity of the advance front curve in the WinSRFR model to soil infiltration parameters is very high. They presented that those simulations results were strongly affected when these parameters were changed. Khorami and Ghahraman [57] solved the Richard equation numerically to simulate soil moisture distribution and soil water infiltration. They reported that the model’s simulation uncertainties are relatively high in terms of moisture and infiltration parameters.

_{0})-NSE; into 0.91 and 1 R lines, 34.90 and 76.30 STD lines, and 1.17 and 31.28 RMSD lines for (k, a, f

_{0})-PBIAS; into 0.91 and 1 R lines, 15.19 and 109.54 STD lines, and 1.17 and 59.85 RMSD lines for (k, a, f

_{0})-R

^{2}; into 0.98 and 1 R lines, 46.69 and 74.95 STD lines, and 7.65 and 25.38 RMSD lines for (n)-NSE; into 0.98 and 1 R lines, 46.69 and 65.60 STD lines, and 7.65 and 16.46 RMSD lines for (n)-PBIAS; into the 0.98 and 1 R lines, 46.69 and 103.37 STD lines, and 7.65 and 53.65 RMSD lines for (n)-R

^{2}. It seems that (SS, BW), (k, a, f

_{0})-NSE and (n)-PBIAS produced more accurate simulations than the others did. However, from Table 5, the estimated advance front curve using geometry cross section properties covers only 50% of observations, while the p-factor is 87.5% for soil infiltration and roughness parameters. The r-factor of estimations after removing non-behavioral data show that the uncertainty width of (k, a, f

_{0})-NSE and (n)-PBIAS reduced from 2.07 to 0.88 and from 0.75 to 0.39, respectively. These results represent that applying the PBIAS function for the roughness parameter of the Manning equation could efficiently reduce uncertain and unstable estimations of the open-ended advance front curve. These uncertainty results indicate an interaction between the accurate measurement of parameters and the output of the WinSRFR model. Nie et al. [58] represented that the performance of the WinSRFR model to simulate water advance and recession trajectories processes are affected by variations in soil infiltration. Moravejalahkamis [59] reported that using a constant value for Manning’s roughness coefficient, soil infiltration properties, field dimensions, inflow rates, and cut-off times in furrow irrigation systems can led to significant errors in advanced recession trajectories assessment. Additionally, several studies suggested that irrigation performance can be improved by considering the soil spatial–temporal variability in soil infiltration and Manning’s roughness parameters [24,31,60]. This statement is in agreement with the results of the current study for the accurate measurement of soil infiltration and Manning’s roughness parameters to achieve the high performance and low uncertainty of the WinSRFR model for simulating the advance front curve.

^{2}> 0.6 resulted in 0.89 < R < 0.94, 12.69 < STD < 27.40, and 5.89 < RMSD < 14.09. The simulation of the flow depth hydrograph using infiltration parameters of (k, a, f

_{0}) resulted in 0.76 < R < 0.99, 10.76 < STD < 22.99, and 2.59 < RMSD < 10.50 for the NSE likelihood function; 0.40 < R < 0.99, 8.56 < STD < 32.84, and 2.59 < RMSD < 22.14 for the PBIAS likelihood function; 0.78 < R < 0.99, 8.56 < STD < 35.22, and 2.59 < RMSD < 24.90 for the R

^{2}likelihood function. By simulating the flow depth hydrograph using the roughness parameter of Manning equation, the minimum and maximum values of R were calculated to be equal to 0.78 and 0.96 for NSE, 0.7 and 0.97 for PBIAS, and 0.78 and 0.97 for R

^{2}likelihood functions, respectively. STD values were changed in the ranges of 10.56–19.10, 7.64–23.37, and 10.35–60.03 for NSE, PBIAS, and R

^{2}likelihood functions, respectively. The RMSD values were changed in the ranges of 6.02–9.68, 6.02–11.34, and 6.02–47.49 for NSE, PBIAS, and R

^{2}likelihood functions, respectively. Based on statistical characteristics, the simulation of flow depth hydrograph using (SS, BW)-NSE, (k, a, f

_{0})-NSE, and (n)-NSE in the condition of close-ended furrow irrigation are more accurate than the others are. Consequently, from the results shown in Table 4, among three estimations, (k, a, f

_{0})-NSE resulted in the lowest value of 95PPU and highest value of r-factor, which are equal to 62.5% and 1.06, respectively. Additionally, based on the results in Table 3, it can be concluded that when measurements are dispersed and involve errors, estimations of the flow depth hydrograph using (n)-NSE may reduce uncertainties. The remained behavioral data using NSE > 0.5 for geometry cross section parameters was 4.76%, which shows the instability of SS and BW data due to the measurement errors.

_{0})-NSE, and (n)-NSE had more accurate simulations. The estimations were grouped into R lines of 0.88 and 0.94, STD lines of 15.60 and 19.34, and RMSD lines of 6.66 and 9.21 for (SS, BW)-NSE; into R lines of 0.74 and 0.99, STD lines of 14.33 and 28.82, and RMSD lines of 3.76 and 13.03 for (k, a, f

_{0})-NSE; into R lines of 0.71 and 0.94, STD lines of 14.17 and 20.54, and RMSD lines of 6.88 and 13.33 for (n)-NSE. Although the 95PPU of (SS, BW)-NSE was 77.8%, which is more than the value related to (n)-NSE, which is equal to 52.78% (Table 5), from the results of Table 4, only 6.58% of the simulations were recognized as behavioral ones. (k, a, f

_{0})-NSE resulted in the highest 95PPU value, and 94.45% of the simulated data were bracketed by the 95% estimation uncertainty. In both the open-ended and close-ended conditions, the width of the uncertainty bound reduced significantly (Figure 14, Table 3 and Table 5).

_{0})-NSE and (n)-NSE showed better performances than the other estimations did. Based on the Taylor diagram given in Figure 15, the statistical results of (k, a, f

_{0})-NSE were calculated as 0.83 < R < 0.89, 0.08 < STD < 0.12, and 0.05 < RMSD < 0.07. The results of 0.88 < R < 0.92, 0.141 < STD < 0.148, and 0.06 < RMSD < 0.07 were obtained for (n)-NSE. When considering the percentage of remained behavioral data (Table 4), it can be seen more than 92% of data were eliminated due to NSE < 0.6. Therefore, re-evaluating the results shows that (n)-PBIAS and (n)-R

^{2}reduced the simulating uncertainty of runoff hydrograph while maintaining the 98.56% and 47.58% of the data, respectively. The statistical results of (n)-PBIAS were exhibited to be 0.23 < R < 0.92, 0.06 < STD < 0.16, and 0.138 < RMSD < 0.158. Those results were calculated as 0.77 < R < 0.92, 0.06 < STD < 0.1, and 0.138 < RMSD < 0.153 for (n)-R

^{2}. Using (n)-PBIAS, 91.18% of observation fell inside the estimated runoff hydrograph. This percentage reduced to 70.59% when (n)-R

^{2}was used. The width of uncertainty bound reduced by applying likelihood criteria of −0.25 < PBIAS < 0.25 and R

^{2}> 0.6 (Figure 16). Regarding Figure 16 and Table 5, the majority of runoff hydrograph observations were bracketed in the estimated bound by using the PBAIS function. It seems the PBAIS function is able to represent the changes in runoff over time. Some observation points of runoff were beyond the 95PPU estimated by the R

^{2}function. Thus, R

^{2}was recognized as a modest likelihood function for the runoff hydrograph.

## 4. Conclusions

_{0}), and Manning’s roughness (n) parameters in different conditions of furrow irrigation system (close end and open end) were evaluated to analyze the impacts of them on the WinSRFR model outputs. The GLUE analysis was performed by considering three likelihood functions of NSE > 0.5, −0.25 < PBAIS < 0.25, and R

^{2}> 0.6 to assess the uncertainty degree of the model outputs response to the input parameters, including advance front curve, flow depth hydrograph, and runoff hydrograph. The following conclusions were drawn from the results:

- -
- The initial evaluation without filtering non-behavioral estimations showed that the model outputs have high uncertainties in regard to the input parameter of geometry cross section. The value width range of uncertainty bound could be reduced by employing Kostiakov–Lewis infiltration and Manning’s roughness parameters.
- -
- Likelihood measures greatly affect the uncertainty bounds, especially with respect to the Kostiakov–Lewis infiltration and Manning’s roughness parameters. The confidence interval and uncertainty bound of the advance front curve and runoff hydrograph did not change in response to employing likelihood functions for geometry cross section parameters.
- -
- There is a higher level of instability in the model outputs related to soil infiltration parameters than those related to the roughness coefficient. This result is related to the higher level of sensitivity of the WinSRFR model in inaccurate measurements, methods, and equipment to monitor the soil infiltration than that which is considering for the optimum roughness coefficient.
- -
- The PBIAS likelihood function played a critical role in the uncertainty results for the estimated advance front curve and runoff hydrograph. In contrast, the NSE likelihood function was more important to implicitly determine the flow depth hydrograph estimation uncertainty. These functions reduced the uncertainty of model outputs by avoiding the parameter sets that produced outputs there were very different from the observations.

^{2}, and PBIAS, (2) evaluating limited input parameters, and (3) investigating one source of model uncertainty, which indicate a need for further research. Thus, finding a robust likelihood function with different threshold values is critical because it strongly influences the width of the uncertainty bounds. Additionally, the influence of the length of furrow, inflow, cut-off time, etc., on the model output uncertainty is unknown when one is applying GLUE for WinSRFR-based furrow irrigation applications. Then, future research can consider other epistemic uncertainties such as model structure and more input parameters. This work can also be replicated by implementing different confidence intervals instead of 2.5% and 97.5%. This study provides a conscious understanding of the correlation between the factors affecting furrow irrigation performance. The methodology of this study can be used for complementary studies on the accuracy evaluation of different models of furrow irrigation system simulation. Then, engineers can make more realistic decisions under uncertainty based on the confidence interval bound, which can help to avoid decisions based on incomplete value of models’ prediction.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The uncertainty bounds of advance front curve for different furrow irrigation conditions and parameters.

**Figure 2.**The uncertainty bounds of flow depth hydrograph for different furrow irrigation conditions and parameters.

**Figure 3.**The uncertainty bounds of runoff hydrograph for open-ended furrow irrigation conditions and different parameters.

**Figure 4.**Posterior (red line) and prior (black line) cumulative distributions of the advance front model from close-ended irrigation conditions.

**Figure 5.**Posterior (red line) and prior (black line) cumulative distributions of the advance front model from open-ended irrigation conditions.

**Figure 6.**Posterior (red line) and prior (black line) cumulative distributions of the flow depth hydrograph model from close-ended irrigation conditions.

**Figure 7.**Posterior (red line) and prior (black line) cumulative distributions of the flow depth hydrograph model from open-ended irrigation conditions.

**Figure 8.**Posterior (red line) and prior (black line) cumulative distributions of the runoff hydrograph model from open-ended irrigation conditions.

**Figure 9.**The Taylor diagrams of WinSRFR advance front outputs for close-ended furrow irrigation condition after removing non-behavioral data.

**Figure 10.**Taylor diagrams of WinSRFR advance front outputs for open-ended condition furrow irrigation after applying behavioral data.

**Figure 11.**The uncertainty bounds of advance front curve for different furrow irrigation conditions after removing non-behavioral data.

**Figure 12.**Taylor diagrams of WinSRFR flow depth hydrograph outputs in the close-ended condition furrow irrigation after applying behavioral data.

**Figure 13.**Taylor diagrams of WinSRFR flow depth hydrograph outputs in the open-ended condition furrow irrigation after applying behavioral data.

**Figure 14.**The uncertainty bounds of flow depth hydrograph for different furrow irrigation conditions after removing non-behavioral data.

**Figure 15.**Taylor diagrams of WinSRFR runoff hydrograph outputs for open-ended condition furrow irrigation after applying behavioral data.

**Figure 16.**The uncertainty bounds of runoff hydrograph for different furrow irrigation conditions after removing non-behavioral data.

Geometry and Field Characteristics | Furrow 1 | Furrow 2 |
---|---|---|

Length (m) | 168.60 | 168.60 |

Downstream condition | Open ended | Closed ended |

Average slope | 2.40 × 10^{−4} | 2.60 × 10^{−4} |

Bottom width (m) | 0.16 | 0.27 |

Side slope | 2.00 | 1.56 |

Manning coefficient | 0.05 | 0.05 |

${Q}_{avg}({\mathrm{m}}^{3}\xb7{\mathrm{s}}^{-1})$ | 1.50 × 10^{−3} | 1.70 × 10^{−3} |

Parameter | Distribution | p-Value | Coefficients | Formulation |
---|---|---|---|---|

BW | Wakeby | 0.13 | $\alpha =1356.70$ $\beta =126.02$ $\gamma =47.50$ $\delta =0.13$ $\xi =0$ | $f\left(x\right)=\xi +\left(\frac{\alpha}{\beta}\right)\left(1-{\left(1-x\right)}^{\beta}\right)-\left(\frac{\gamma}{\delta}\right)\left(1-{\left(1-x\right)}^{-\delta}\right)$ |

SS | Wakeby | 0.13 | $\alpha =54.20$ $\beta =63.10$ $\gamma =0.57$ $\delta =-0.15$ $\xi =0$ | |

k | Log-Logistic | 0.12 | $\alpha =2.50$ $\beta =24.9$ | $f\left(x\right)=\frac{\left(\frac{\alpha}{\beta}\right){\left(\frac{x}{\beta}\right)}^{\alpha -1}}{{\left(1+{\left(\frac{x}{\beta}\right)}^{\alpha}\right)}^{2}}$ |

a | Wakeby | 0.12 | $\alpha =1.50$ $\beta =2.60$ $\gamma =0.01$ $\delta =0.40$ $\xi =-0.01$ | |

f_{0} | Wakeby | 0.12 | $\alpha =16.40$ $\beta =2.40$ $\gamma =6.90$ $\delta =0.03$ $\xi =-0.90$ | |

n | Wakeby | 0.19 | $\alpha =0.10$ $\beta =5$ $\gamma =0.01$ $\delta =0.50$ $\xi =0.006$ |

Conditions | Uncertainty Criteria | Advance Front | Depth Hydrograph | Flow Hydrograph |
---|---|---|---|---|

Close end (SS, BW) | p-factor | 12.50 | 34.38 | |

r-factor | 0.11 | 4.40 | ||

Open end (SS, BW) | p-factor | 50 | 52.78 | 14.71 |

r-factor | 0.11 | 3.44 | 0.48 | |

Close end (k, a, f_{0}) | p-factor | 100 | 100 | |

r-factor | 1.14 | 3.99 | ||

Open end (k, a, f_{0}) | p-factor | 87.50 | 100 | 100 |

r-factor | 2.07 | 2.32 | 11.49 | |

Close end (n) | p-factor | 87.50 | 100 | |

r-factor | 0.64 | 3.92 | ||

Open end (n) | p-factor | 87.50 | 100 | 97.06 |

r-factor | 0.75 | 3.52 | 2.07 |

Conditions | Likelihood Criteria | Advance Front | Depth Hydrograph | Flow Hydrograph |
---|---|---|---|---|

Close end (SS, BW) | NSE | 100 | 4.76 | |

R^{2} | 100 | 100 | ||

PBIAS | 100 | 7.69 | ||

Open end (SS, BW) | NSE | 100 | 6.58 | 0 |

R^{2} | 100 | 100 | 100 | |

PBIAS | 100 | 9.01 | 100 | |

Close end (k, a, f_{0}) | NSE | 56.35 | 14.06 | |

R^{2} | 76.08 | 57.37 | ||

PBIAS | 21.54 | 48.75 | ||

Open end (k, a, f_{0}) | NSE | 53.40 | 15.76 | 0.84 |

R^{2} | 81.41 | 81.41 | 59.75 | |

PBIAS | 23.36 | 35.71 | 6.41 | |

Close end (n) | NSE | 96.61 | 50.67 | |

R^{2} | 96.61 | 68.40 | ||

PBIAS | 86.64 | 76.10 | ||

Open end (n) | NSE | 96.20 | 50.05 | 7.61 |

R^{2} | 100 | 64.75 | 47.58 | |

PBIAS | 89.41 | 72.05 | 98.56 |

Conditions | Criteria | Advance Front | Depth Hydrograph | Flow Hydrograph | ||||||
---|---|---|---|---|---|---|---|---|---|---|

NSE | PBIAS | R^{2} | NSE | PBIAS | R^{2} | NSE | PBIAS | R^{2} | ||

Close end (SS, BW) | p-factor | 12.5 | 12.5 | 12.5 | 87.5 | 90.6 | 34.38 | |||

r-factor | 0.11 | 0.11 | 0.11 | 0.99 | 1.42 | 4.40 | ||||

Open end (SS, BW) | p-factor | 50 | 50 | 50 | 77.8 | 80.56 | 52.78 | 0 | 14.71 | 14.71 |

r-factor | 0.11 | 0.11 | 0.11 | 1.05 | 1.21 | 3.44 | 0 | 0.48 | 0.48 | |

Close end (k, a, f_{0}) | p-factor | 87.5 | 87.5 | 87.5 | 62.5 | 100 | 100 | |||

r-factor | 0.63 | 0.51 | 0.70 | 1.06 | 2.09 | 3.06 | ||||

Open end (k, a, f_{0}) | p-factor | 87.5 | 87.5 | 87.5 | 94.45 | 100 | 100 | 32.37 | 41.18 | 44.12 |

r-factor | 0.88 | 0.66 | 1.44 | 1.32 | 1.85 | 2.24 | 0.47 | 1.89 | 8.98 | |

Close end (n) | p-factor | 87.5 | 87.5 | 87.5 | 71.87 | 90.63 | 90.63 | |||

r-factor | 0.51 | 0.40 | 0.51 | 0.93 | 1.75 | 3.10 | ||||

Open end (n) | p-factor | 87.5 | 87.5 | 87.5 | 52.78 | 72.22 | 61.11 | 29.41 | 91.18 | 70.59 |

r-factor | 0.52 | 0.39 | 0.75 | 0.76 | 1.37 | 2.69 | 0.37 | 2.01 | 1.06 |

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## Share and Cite

**MDPI and ACS Style**

Seifi, A.; Golestani Kermani, S.; Mosavi, A.; Soroush, F.
Uncertainty Assessment of WinSRFR Furrow Irrigation Simulation Model Using the GLUE Framework under Variability in Geometry Cross Section, Infiltration, and Roughness Parameters. *Water* **2023**, *15*, 1250.
https://doi.org/10.3390/w15061250

**AMA Style**

Seifi A, Golestani Kermani S, Mosavi A, Soroush F.
Uncertainty Assessment of WinSRFR Furrow Irrigation Simulation Model Using the GLUE Framework under Variability in Geometry Cross Section, Infiltration, and Roughness Parameters. *Water*. 2023; 15(6):1250.
https://doi.org/10.3390/w15061250

**Chicago/Turabian Style**

Seifi, Akram, Soudabeh Golestani Kermani, Amir Mosavi, and Fatemeh Soroush.
2023. "Uncertainty Assessment of WinSRFR Furrow Irrigation Simulation Model Using the GLUE Framework under Variability in Geometry Cross Section, Infiltration, and Roughness Parameters" *Water* 15, no. 6: 1250.
https://doi.org/10.3390/w15061250