# Evaluating Monthly Flow Prediction Based on SWAT and Support Vector Regression Coupled with Discrete Wavelet Transform

^{1}

^{2}

^{*}

## Abstract

**:**

^{3}s

^{−1}in calibration and 0.14, 0.98, −1.88, and 2.91 m

^{3}s

^{−1}in validation on 12 sites, respectively. Compared with the other two models, the proposed SWAT-WSVR model possessed lower discrepancy and higher accuracy. The rank of the overall performance of the three SWAT-based models during the whole study period was SWAT-WSVR > SWAT-SVR > SWAT-CUP. The developed SWAT-WSVR model supplies an additional calibration approach that can improve the accuracy of the SWAT streamflow simulation of watersheds with limited data.

## 1. Introduction

## 2. Methodology

#### 2.1. Watershed Description and Data Source

^{2}drainage area and crosses Arkansas and Oklahoma, U.S. The average basin slope is about 5.6%. The average annual temperature and precipitation are about 16 °C and 1198 mm, respectively. Monthly statistical discharge data from twelve U.S. Geological Survey (USGS) hydrological sites were downloaded from the official website [21]. These monthly statistics generated from sites are based on USGS-approved daily-mean data. The basic information of 12 hydrologic stations and selected descriptive statistics for monthly flow time series are listed in Table 1. Additionally, Figure 2 shows the spatial distribution of meteorological and hydrological stations, terrain, lakes, and rivers in the IRW.

#### 2.2. Hydrological Model

#### 2.3. Support Vector Machine

^{−6}, end = 2

^{8}, step = 1), γ (begin = 2

^{4}, end = 2

^{−8}, step = −1), and ε (begin = 2

^{−8}, end = 2

^{−1}, step = 0.5). The k-value in cross-validation was set to 5 for tuning the SVR. Before training the SVR, all input data were normalized to the value range [0, 1] by the formula $\left(x-x\_min\right)/\left(x\_max-x\_min\right)$. Additionally, for each site, we used the first 70% of data to train the model, then applied the remaining 30% subset for validation purposes. The SVR ε-regression model was used to develop both SWAT-SVR and SWAT-WSVR. R version 4.1.0 running on RStudio version 1.4.1717 and the ‘e1071’ package [44] were used for the development, training, and testing of the hybrid model [45].

#### 2.4. Wavelet Transforms

#### 2.5. Model Performance Evaluation

## 3. Results and Discussion

#### 3.1. Flow Prediction by SWAT-CUP

^{3}s

^{−1}in calibration and 0.84, 0.26, 35.98, and 11.04 m

^{3}s

^{−1}in validation, respectively. The RSR, NSE, and RMSE had approximately similar performances between calibration and validation, but the value of PBIAS in validation was lower than one in calibration, which indicated that the predicted discrepancy from validation was less than one from calibration. SWAT-CUP overestimated monthly flow in both calibration and validation. The low average NSE value (≤0.26) indicated that SWAT-CUP has a poor goodness-of-fit between the observed and simulated flow for both calibration and validation. Additionally, the 07195430 site had the best performance among all sites in validation with lowest PBIAS (−6.7) and RSR (0.58) and the highest NSE value (0.66). The model performance of 07195500 and 07196500 were also acceptable in validation. After combined calibration and validation data together, the averages of RSR, NSE, PBIAS, and RMSE for 12 sites are 0.85, 0.13, 50.67, and 11.38 m

^{3}s

^{−1}, respectively. Simulations from SWAT-CUP greatly overestimated the observed flow according to PBIAS (i.e., a positive mean of 50.67 for 12 sites) and presented a low fitting degree due to a low average value (0.13) of NSE. Overall, SWAT-CUP had a poor simulation performance for most of sites during both calibration and validation periods, with only few exceptions.

#### 3.2. Flow Prediction by SWAT-SVR

^{3}s

^{−1}in calibration and 0.80, 0.32, −20.68, and 11.30 m

^{3}s

^{−1}in validation (Table 4), respectively. Compared with SWAT-CUP, the performance of the SWAT-SVR model had the lower RSR and the absolute value of PBIAS and the higher NSE in calibration and validation, particularly for the calibrated simulations, as SVR has a strong learning ability for training data. The average RMSE in SWAT-SVR calibration is lower: only 7.34 m

^{3}s

^{−1}compared with the average one of 11.50 m

^{3}s

^{−1}in the SWAT-CUP calibration. The results showed that the SWAT-SVR calibration on all sites had lower deviation and higher NSE value in comparison with SWAT-CUP, but still underestimated monthly flows on most sites. SWAT-SVR was generally superior to SWAT-CUP on all sites. However, only a few sites (e.g., 07195500, 07195430) had lower RSR, PBIAS, and higher NSE values, which indicated that SWAT-SVR could greatly improve the model performance in calibration but did not possess good generalization capability, which means it failed to keep this prediction ability with high accuracy while it was applied in validation. From the perspective of the whole data series, the average RSR, NSE, PBIAS, and RMSE are 0.65, 0.57, −13.03, and 8.84 m

^{3}s

^{−1}, respectively. Clearly, compared with SWAT-CUP, the performance of the SWAT-SVR model were improved but limited, although SWAM-SVR generally had a low RSR, absolute value of PBIAS, and higher NSE value for calibration, validation, and both periods.

#### 3.3. SWAT-WSVR Development and Evaluation

#### 3.3.1. Development of SWAT-WSVR

#### 3.3.2. Statistical Evaluation of SWAT-WSVR

^{3}s

^{−1}in calibration; 0.14, 0.98, −1.88, and 2.91 m

^{3}s

^{−1}in validation; and 0.08, 0.99, −0.74, and 1.61 m

^{3}s

^{−1}in the whole data series. Compared with SWAT-SVR and SWAT-CUP, SWAT-WSVR had the lowest RSR, the absolute value of PBIAS, and RMSE but the highest NSE value in validation. This result clearly indicated that SWAT-WSVR could effectively decrease the discrepancy of the simulation and obtain the best prediction accuracy for validation in comparison with SWAT-SVR and SWAT-CUP. Based on the value of PBIAS, SWAT-WSVR slightly underestimated the monthly flow in calibration. SWAT-WSVR also presented the best performance on the whole data series, along with the lowest RSR, PBIAS, and RMSE but the highest NSE in comparison with SWAT-CUP and SWAT-SVR. By comparison, the SWAT-WSVR model outperformed the SWAT-CUP and SWAT-SVR model in calibration, validation, and both periods.

#### 3.4. Taylor Diagram and Hydrographic Comparison between Different Models

^{3}s

^{−1}) was 27.2% and 27.9% lower than observed flow (144.61 and 149.42 m

^{3}s

^{−1}) in April 2011 at 07195430 and 07195500 site, respectively. Overall, the other two models more or less capture the rising and recession of the observed monthly flow over time at all sites, but the SWAT-WSVR is more efficient at fitting with the observation and corrected errors compared to SWAT-CUP. This result is in line with others’ conclusions that the application of wavelet transform in data-driven models can improve the accuracy of flow prediction [53,54]. The developed SWAT-WSVR model fit the observations well at all sites of the IRW based on the statistical results, Taylor diagram, and hydrography analysis.

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**The geographic distribution of NCDC meteorological stations, USGS hydrological stations, lakes, and rivers in the IRW.

**Figure 4.**An example of the flow DWT at 07195430 site: (

**a**) Observation and SWAT-CUP simulation, (

**b**) wavelet decomposition.

**Figure 5.**Correlation coefficient matrices of input variables in the SWAT-WSVR model (Note: ‘Prec’ denotes precipitation).

**Figure 7.**Comparison of SWAT-CUP, SWAT-SVR, SWAT-WSVR, and observed flow time series on each site. (Note: a vertical blue line separated the 70% training and 30% testing data. Only statistics from SWAT-WSVR were labeled for clarity).

No. | USGS Station | Upstream Area (km ^{2}) | Data Period (month.year) | Number of Data | Average Monthly Streamflow (m ^{3} s^{−1}) | Flow Descriptive Statistics (m^{3} s^{−1}) | |||
---|---|---|---|---|---|---|---|---|---|

Max | Min | Median | Standard Deviation | ||||||

1 | 07195800 | 36.8 | 1.1995–12.2013 | 228 | 0.41 | 2.90 | 0.05 | 0.25 | 0.44 |

2 | 07195855 | 155.0 | 1.1995–12.2013 | 228 | 1.27 | 9.53 | 0.11 | 0.74 | 1.43 |

3 | 07196000 | 300.7 | 1.1995–12.2013 | 228 | 3.01 | 22.26 | 0.42 | 1.78 | 3.25 |

4 | 07195500 | 1633.0 | 1.1995–12.2013 | 228 | 18.71 | 149.42 | 2.73 | 10.75 | 20.02 |

5 | 07195430 | 1490.5 | 1.1996–12.2013 | 216 | 17.68 | 144.61 | 1.89 | 10.58 | 19.29 |

6 | 07196090 | 2138.5 | 7.2010–12.2013 | 42 | 23.19 | 178.54 | 2.95 | 11.77 | 33.59 |

7 | 07196973 | 64.8 | 1.1995–12.2002 | 96 | 0.66 | 3.57 | 0.00 | 0.38 | 0.75 |

8 | 07196500 | 2462.5 | 1.1995–12.2013 | 228 | 27.76 | 190.80 | 2.99 | 17.09 | 30.17 |

9 | 07197000 | 808.7 | 1.1995–12.2013 | 228 | 9.27 | 69.73 | 0.33 | 4.93 | 11.44 |

10 | 07196900 | 105.2 | 1.1995–12.2013 | 228 | 1.31 | 10.35 | 0.00 | 0.59 | 1.76 |

11 | 07197360 | 233.8 | 1.1998–12.2013 | 192 | 2.41 | 15.18 | 0.10 | 1.46 | 2.88 |

12 | 07198000 | 4186.2 | 1.1995–12.2013 | 228 | 44.03 | 378.65 | 0.98 | 25.59 | 46.81 |

No. | Parameter Name ^{†} | Parameter Description | Range | Fitted Value |
---|---|---|---|---|

1 | R__CN2.mgt | SCS runoff curve number II | −0.25–0.25 | −0.179 |

2 | V__GWQMN.gw | Threshold depth of water in the shallow aquifer required for return flow to occur (mm H_{2}O) | 0–2000 | 1764 |

3 | V__GW_REVAP.gw | Groundwater “revap” coefficient | 0.02–0.2 | 0.135 |

4 | V__REVAPMN.gw | Threshold depth of water in the shallow aquifer for ‘revap’ to occur (mm) | 0–500 | 121 |

5 | V__EPCO.hru | Plant uptake compensation factor | 0–1 | 0.154 |

6 | V_ESCO.hru | Soil evaporation compensation factor | 0–1 | 0.354 |

7 | R__SOL_AWC (1).sol | Available water capacity of the 1st soil layer (mm H_{2}O mm soil^{−1}) | 0.08–0.2 | 0.177 |

8 | A__OV_N.hru | Manning’s “n” value for overland flow | 0.01–30 | 26.941 |

9 | R__HRU_SLP.hru | Average slope steepness (m m^{−1}) | 0–1 | 0.034 |

Indicator Name | Calculation Equation ^{†} | Description |
---|---|---|

Pearson’s Correlation Coefficient (r) | $r=\frac{n({\displaystyle \sum}y{y}_{i})-\left({\displaystyle \sum}y\right)\left({\displaystyle \sum}{y}_{i}\right)}{\sqrt{\left[n{\displaystyle \sum}{y}^{2}-{\left({\displaystyle \sum}y\right)}^{2}\right]\left[n{\displaystyle \sum}{y}^{\prime}-{\left({\displaystyle \sum}{y}^{\prime}\right)}^{2}\right]}}$ | Range [−1, 1] |

Nash–Sutcliffe efficiency (NSE) | $NSE=1-\frac{{{\displaystyle \sum}}_{i=1}^{n}{({y}_{i}-{y}_{i}^{\prime})}^{2}}{{{\displaystyle \sum}}_{i=1}^{n}{({y}_{i}-\overline{y})}^{2}}$ | Range (−∞, 1], and 1 is the optimal value |

Percent Bias (PBIAS) | $PBIAS=100\times \frac{{{\displaystyle \sum}}_{i=1}^{n}\left({y}_{i}{}^{\prime}-{y}_{i}\right)}{{{\displaystyle \sum}}_{i=1}^{n}{y}_{i}}$ | Range (−∞, +∞), and 0 is the optimal value |

RMSE-observations standard deviation ratio (RSR) | $RSR=\frac{\sqrt{{{\displaystyle \sum}}_{i=1}^{n}{({y}_{i}-{y}_{i}{}^{\prime})}^{2}}}{\sqrt{{{\displaystyle \sum}}_{i=1}^{n}{({y}_{i}-\overline{y})}^{2}}}$ | Range [0, +∞), and 0 is the optimal value |

Root Mean Square Error (RMSE) | $\mathit{RMSE}=\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{n}{\left({y}_{i}-{y}_{i}^{\prime}\right)}^{2}}{n}}$ | Range [0, +∞), and 0 is the optimal value |

**Table 4.**Performance of flow calibration, validation, and combined data series on each site by SWAT-CUP, SWAT-SVR, and SWAT-WSVR.

Station | SWAT-CUP | SWAT-SVR | SWAT-WSVR | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

RSR | NSE | PBIAS | RMSE (m ^{3} s^{−1}) | RSR | NSE | PBIAS | RMSE (m ^{3} s^{−1}) | RSR | NSE | PBIAS | RMSE (m ^{3} s^{−1}) | ||

Calibration | 07195800 | 0.76 | 0.41 | 24.00 | 0.33 | 0.58 | 0.68 | −10.80 | 0.25 | 0.08 | 0.99 | −1.90 | 0.04 |

07195855 | 0.95 | 0.09 | 66.50 | 1.32 | 0.63 | 0.60 | −10.00 | 0.88 | 0.01 | 1.00 | 0.10 | 0.01 | |

07196000 | 0.96 | 0.07 | 67.90 | 3.18 | 0.57 | 0.68 | 0.50 | 1.88 | 0.01 | 1.00 | 0.10 | 0.02 | |

07195500 | 0.74 | 0.46 | 44.60 | 13.68 | 0.55 | 0.70 | −5.50 | 10.23 | 0.01 | 1.00 | 0.10 | 0.23 | |

07195430 | 0.67 | 0.54 | 29.30 | 11.40 | 0.55 | 0.70 | −5.30 | 9.30 | 0.01 | 1.00 | 0.20 | 0.22 | |

07196090 | 0.52 | 0.72 | 31.40 | 19.81 | 0.14 | 0.98 | −3.30 | 5.18 | 0.02 | 1.00 | −0.60 | 0.70 | |

07196973 | 10.70 | −0.15 | 69.90 | 0.78 | 0.73 | 0.46 | −6.50 | 0.53 | 0.02 | 1.00 | 0.20 | 0.01 | |

07196500 | 0.78 | 0.39 | 52.30 | 22.62 | 0.58 | 0.67 | −10.20 | 16.71 | 0.02 | 1.00 | 0.10 | 0.48 | |

07197000 | 0.87 | 0.24 | 69.50 | 10.04 | 0.57 | 0.67 | −11.80 | 6.64 | 0.01 | 1.00 | 0.10 | 0.09 | |

07196900 | 0.93 | 0.14 | 86.00 | 1.69 | 0.57 | 0.68 | −13.60 | 1.04 | 0.01 | 1.00 | 0.00 | 0.02 | |

07197360 | 0.97 | 0.06 | 74.90 | 2.89 | 0.70 | 0.51 | −21.30 | 2.07 | 0.00 | 1.00 | 0.00 | 0.01 | |

07198000 | 1.17 | −0.39 | 74.50 | 50.21 | 0.78 | 0.39 | −15.90 | 33.42 | 0.03 | 1.00 | −0.20 | 1.41 | |

Mean | 1.67 | 0.22 | 57.57 | 11.50 | 0.58 | 0.64 | −9.48 | 7.34 | 0.02 | 1.00 | −0.15 | 0.27 | |

Validation | 07195800 | 0.85 | 0.26 | 8.70 | 0.35 | 0.89 | 0.20 | −20.40 | 0.36 | 0.11 | 0.99 | −2.90 | 0.04 |

07195855 | 0.89 | 0.19 | 22.80 | 1.35 | 0.81 | 0.33 | −27.00 | 1.23 | 0.09 | 0.99 | −1.10 | 0.14 | |

07196000 | 0.98 | 0.03 | 29.60 | 3.03 | 0.86 | 0.24 | −16.00 | 2.68 | 0.14 | 0.98 | −1.00 | 0.43 | |

07195500 | 0.59 | 0.65 | 15.40 | 13.59 | 0.58 | 0.65 | −20.50 | 13.48 | 0.24 | 0.94 | −4.60 | 5.49 | |

07195430 | 0.58 | 0.66 | −6.70 | 13.88 | 0.58 | 0.68 | −23.80 | 13.74 | 0.24 | 0.94 | −4.80 | 5.77 | |

07196090 | 0.71 | 0.45 | 38.30 | 14.32 | 1.15 | −0.43 | −41.70 | 23.16 | 0.12 | 0.98 | 0.70 | 2.47 | |

07196973 | 1.03 | −0.10 | 62.30 | 0.83 | 0.87 | 0.22 | −15.70 | 0.69 | 0.06 | 1.00 | 0.70 | 0.05 | |

07196500 | 0.66 | 0.56 | 19.70 | 21.55 | 0.63 | 0.60 | −25.50 | 20.62 | 0.13 | 0.98 | −2.20 | 4.28 | |

07197000 | 0.88 | 0.22 | 64.80 | 9.84 | 0.72 | 0.47 | −6.30 | 8.07 | 0.10 | 0.99 | −0.40 | 1.17 | |

07196900 | 1.07 | −0.17 | 87.90 | 1.72 | 0.96 | 0.06 | 3.60 | 1.54 | 0.05 | 0.99 | 0.60 | 0.08 | |

07197360 | 0.88 | 0.21 | 55.80 | 2.34 | 0.66 | 0.58 | −25.10 | 1.76 | 0.07 | 0.99 | −1.40 | 0.18 | |

07198000 | 0.90 | 0.18 | 33.20 | 49.62 | 0.84 | 0.29 | −29.80 | 48.22 | 0.27 | 0.93 | −6.10 | 14.81 | |

Mean | 0.84 | 0.26 | 35.98 | 11.04 | 0.80 | 0.32 | −20.68 | 11.30 | 0.14 | 0.98 | −1.88 | 2.91 | |

The whole series data ^{†} | 07195800 | 0.79 | 0.37 | 19.50 | 0.33 | 0.68 | 0.53 | −13.60 | 0.29 | 0.09 | 0.99 | −2.20 | 0.04 |

07195855 | 0.93 | 0.13 | 52.30 | 1.33 | 0.70 | 0.51 | −15.50 | 1.00 | 0.06 | 1.00 | −0.30 | 0.08 | |

07196000 | 0.97 | 0.06 | 55.90 | 3.13 | 0.66 | 0.56 | −4.60 | 2.15 | 0.07 | 0.99 | −0.30 | 0.24 | |

07195500 | 0.68 | 0.53 | 35.00 | 13.65 | 0.56 | 0.68 | −10.40 | 11.30 | 0.15 | 0.98 | −1.50 | 3.00 | |

07195430 | 0.63 | 0.60 | 16.80 | 12.19 | 0.56 | 0.68 | −11.70 | 10.81 | 0.16 | 0.97 | −1.60 | 3.15 | |

07196090 | 0.55 | 0.69 | 33.50 | 18.41 | 0.39 | 0.84 | −14.70 | 13.13 | 0.04 | 1.00 | −0.30 | 1.45 | |

07196973 | 1.06 | −0.14 | 67.80 | 0.79 | 0.78 | 0.38 | −9.10 | 0.58 | 0.04 | 1.00 | 0.30 | 0.03 | |

07196500 | 0.74 | 0.45 | 41.80 | 22.31 | 0.60 | 0.64 | −15.10 | 17.97 | 0.08 | 0.99 | −0.60 | 2.36 | |

07197000 | 0.87 | 0.24 | 68.20 | 9.98 | 0.62 | 0.61 | −10.30 | 7.90 | 0.06 | 1.00 | −0.10 | 0.64 | |

07196900 | 0.97 | 0.06 | 86.50 | 1.70 | 0.69 | 0.53 | −8.70 | 1.21 | 0.03 | 1.00 | 0.20 | 0.05 | |

07197360 | 0.95 | 0.10 | 69.50 | 2.73 | 0.69 | 0.52 | −22.40 | 1.98 | 0.03 | 1.00 | −0.40 | 0.10 | |

07198000 | 1.07 | −1.50 | 61.20 | 50.03 | 0.81 | 0.35 | −20.30 | 37.70 | 0.17 | 0.97 | −2.10 | 8.17 | |

Mean | 0.85 | 0.13 | 50.67 | 11.38 | 0.65 | 0.57 | −13.03 | 8.84 | 0.08 | 0.99 | −0.74 | 1.61 |

^{†}Note: The data series combined calibration and validation time series.

Station | SWAT-SVR | SWAT-WSVR | |||||
---|---|---|---|---|---|---|---|

Model Input ^{†} | C | γ | Model Input | Decomposition Levels | C | γ | |

07195800 | Flow + Prec | 36.015625 | 3 | Prec + D1 + D2 + D3 + A3 | 3 | 5.015625 | 1 |

07195855 | Flow + Prec | 22.015625 | 1 | Prec + D1 + D2 + D3 + A3 | 3 | 2.015625 | 3 |

07196000 | Flow + Prec | 255.015625 | 1 | Prec + D1 + D2 + D3 + A3 | 3 | 255.015625 | 1 |

07195500 | Flow + Prec | 103.015625 | 1 | Prec + D1 + D2 + D3 + A3 | 3 | 96.015625 | 1 |

07195430 | Flow + Prec | 255.015625 | 1 | Prec + D1 + D2 + D3 + A3 | 3 | 87.015625 | 1 |

07196090 | Flow + Prec | 255.015625 | 5 | Prec + D1 + D2 + A2 | 2 | 255.015625 | 1 |

07196973 | Flow + Prec | 2.015625 | 1 | Prec + D1 + D2 + A2 | 2 | 5.015625 | 1 |

07196500 | Flow + Prec | 130.015625 | 1 | Prec + D1 + D2 + D3 + A3 | 3 | 125.015625 | 1 |

07197000 | Flow + Prec | 57.015625 | 1 | Prec + D1 + D2 + D3 + A3 | 3 | 242.015625 | 1 |

07196900 | Flow + Prec | 4.015625 | 14 | Prec + D1 + D2 + D3 + A3 | 3 | 15.015625 | 1 |

07197360 | Flow + Prec | 13.015625 | 1 | Prec + D1 + D2 + D3 + A3 | 3 | 35.015625 | 1 |

^{†}Note: Flow comes from SWAT-CUP simulated discharge output. Ds and As are wavelet components from the simulated flow of SWAT-CUP.

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## Share and Cite

**MDPI and ACS Style**

Yuan, L.; Forshay, K.J. Evaluating Monthly Flow Prediction Based on SWAT and Support Vector Regression Coupled with Discrete Wavelet Transform. *Water* **2022**, *14*, 2649.
https://doi.org/10.3390/w14172649

**AMA Style**

Yuan L, Forshay KJ. Evaluating Monthly Flow Prediction Based on SWAT and Support Vector Regression Coupled with Discrete Wavelet Transform. *Water*. 2022; 14(17):2649.
https://doi.org/10.3390/w14172649

**Chicago/Turabian Style**

Yuan, Lifeng, and Kenneth J. Forshay. 2022. "Evaluating Monthly Flow Prediction Based on SWAT and Support Vector Regression Coupled with Discrete Wavelet Transform" *Water* 14, no. 17: 2649.
https://doi.org/10.3390/w14172649