# A Comparative Study of Groundwater Level Forecasting Using Data-Driven Models Based on Ensemble Empirical Mode Decomposition

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Ensemble Empirical Mode Decomposition

- Step 1. Identify all local maxima and minima points of the time series $x(t)$;
- Step 2. Interpolate between all local maxima and minima points of $x(t)$ to form the upper envelope ${e}_{\mathrm{max}}(t)$ and lower envelope ${e}_{\mathrm{min}}(t)$;
- Step 3. Compute the mean envelope $m(t)$ between the upper envelope ${e}_{\mathrm{max}}(t)$ and the lower envelope ${e}_{\mathrm{min}}(t)$;$$m(t)=\left({e}_{\mathrm{max}}(t)+{e}_{\mathrm{min}}(t)\right)/2$$
- Step 4. Calculate the IMF candidate;$$h(t+1)=x(t)-m(t)$$
- Step 5. Determine whether or not $h(t+1)$ satisfies the two conditions of IMF. Is $h(t+1)$ an IMF?

- Step 6. Continue until the final residue meets some predefined stopping criteria.

- Step 1. Initialize the ensemble number and the amplitude of the added white noise.
- Step 2. Add random white noise to produce the noise-added data.
- Step 3. Identify the local maxima and minima and obtain the upper and lower envelopes.
- Step 4. Compute the mean of the upper and lower envelopes.
- Step 5. Decompose the data with added random white noise into IMFs.
- Step 6. Repeat step 3 to step 5 until the stopping criteria. After the shift processing, the IMFs and the residue are obtained.

#### 2.2. Artificial Neural Network

#### 2.3. Support Vector Machine

#### 2.4. Adaptive Neuro Fuzzy Inference System

_{1}and y is B

_{1}; then f

_{1}= p

_{1}x + q

_{1}y + r

_{1}

_{2}and y is B

_{2}; then f

_{2}= p

_{2}x + q

_{2}y + r

_{2}

#### 2.5. The Hybrid EEMD-ANN, EEMD-SVM and EEMD-ANFIS Forecasting Models

- (1)
- Firstly, use the EEMD technique to decompose the original groundwater level fluctuation time series $x\left(t\right)$$\left(t=1,2,\cdots ,n;i=1,2,\cdots ,m\right)$ into an IMF component ${c}_{i}\left(t\right)$ and one residual component ${r}_{m}\left(t\right)$.
- (2)
- Secondly, the three data driven models ANN, SVM and ANFIS are developed to make the corresponding prediction using extracted IMF component and the residual component, respectively.
- (3)
- Finally, all of the predicting results from the ANN, SVM and ANFIS models are combined to obtain the new output values, which are the final forecasting result for the groundwater level prediction.

## 3. Study Area and Available Data

#### 3.1. Study Site and Data Preprocessing

#### 3.2. Performance Criteria

## 4. Decomposing, Modeling and Application

## 5. Results and Discussion

#### 5.1. EEMD-ANN and ANN Models

#### 5.2. EEMD-SVM and SVM Models

#### 5.3. EEMD-ANFIS and ANFIS Models

#### 5.4. Comparison of EEMD-ANN, EEMD-SVM and EEMD-ANFIS

^{2}) values, corresponding to the predicted values in the scatter plots at the M1255 and STL185 observation wells, using the EEMD-ANN, EEMD-SVM, EEMD-ANFIS, ANN, SVM, ANFIS, ANN+L, SVM+L and ANFIS+L models. The scatter plots revealed the relationships between the predicted and observed groundwater levels for two observation wells. It can be seen clearly from the scatter plots that the EEMD-ANFIS model forecast the groundwater levels with less scatter for the two observed wells. Figure 14 and Figure 15 show that EEMD-ANFIS had the best fit line compared to the other models. Figure 16a,b show the forecast groundwater levels versus observed groundwater levels using all of the models in the training stage and the validation stage.

^{2}value for the data at site M1255 indicated that EEMD-ANFIS performed better than the other models, although the RMSE value for the EEMD-SVM was less than that for EEMD-ANFIS. The R

^{2}value of the three hybrid models (i.e., EEMD coupled) were better than that of the other models. Therefore, both the EEMD-ANFIS and EEMD-SVM models can be considered good data-driven models at site M1255. Figure 15 shows that the R

^{2}value of EEMD-ANFIS at site STL185 was nearly equal to that of ANFIS+L and was a bit higher than that of EEMD-ANN. The RMSE value for EEMD-ANFIS was less than that for EEMD-ANN and ANFIS+L. Thus, the EEMD-ANFIS model can be considered the best estimation model at site STL185. For the two observation sites, the forecast results obtained from the three hybrid models were suggested to have better quality compared to those not coupled with EEMD.

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**Location map of the United States and observation wells (Source: Imagery © Data SIO, NOAA, U.S. Navy, NGA, GEBCO, Image Landsat/Copernicus).

**Figure 8.**The partial autocorrelation function (PACF) of the groundwater level series at site M1255.

**Figure 11.**Observed and predicted groundwater level using ANN, ANN+L and EEMD-ANN (

**a**) M1255 (

**b**) STL185.

**Figure 12.**Observed and predicted groundwater levels using SVM, SVM+L and EEMD-SVM (

**a**) M1255 (

**b**) STL185.

**Figure 13.**Observed and predicted groundwater levels using ANFIS, ANFIS+L and EEMD-ANFIS (

**a**) M1255 (

**b**) STL185.

**Figure 14.**Observed and predicted groundwater levels at the M1255 observation well using (

**a**) EEMD-ANN; (

**b**) ANN; (

**c**) ANN+L; (

**d**) EEMD-SVM; (

**e**) SVM; (

**f**) SVM+L; (

**g**) EEMD-ANFIS; (

**h**) ANFIS; (

**i**) ANFIS+L. ($x$ is observed groundwater level, $y$ is predicted groundwater level).

**Figure 15.**Observed and predicted groundwater levels at the STL185 observation well using (

**a**) EEMD-ANN; (

**b**) ANN; (

**c**) ANN+L; (

**d**) EEMD-SVM; (

**e**) SVM; (

**f**) SVM+L; (

**g**) EEMD-ANFIS; (

**h**) ANFIS; (

**i**) ANFIS+L. ($x$ is observed groundwater level, $y$ is predicted groundwater level).

Training | Validation | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

R | NMSE | RMSE | NS | AIC | R | NMSE | RMSE | NS | AIC | ||

EEMD-ANN | 0.932 | 0.131 | 0.234 | 0.869 | −488.040 | 0.926 | 0.317 | 0.329 | 0.669 | −53.350 | |

M1255 | ANN | 0.808 | 0.345 | 0.380 | 0.653 | −325.009 | 0.737 | 0.442 | 0.389 | 0.538 | −45.377 |

ANN+L | 0.816 | 0.333 | 0.374 | 0.665 | −330.668 | 0.754 | 0.675 | 0.480 | 0.296 | −35.246 | |

EEMD-ANN | 0.977 | 0.044 | 0.281 | 0.955 | −426.784 | 0.891 | 0.208 | 0.646 | 0.783 | −20.979 | |

STL185 | ANN | 0.869 | 0.244 | 0.659 | 0.754 | −140.343 | 0.813 | 0.353 | 0.842 | 0.632 | −8.239 |

ANN+L | 0.872 | 0.239 | 0.651 | 0.760 | −144.026 | 0.839 | 0.298 | 0.774 | 0.689 | −12.323 |

Training | Validation | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

R | NMSE | RMSE | NS | AIC | R | NMSE | RMSE | NS | AIC | ||

EEMD-SVM | 0.948 | 0.102 | 0.206 | 0.898 | −530.463 | 0.885 | 0.292 | 0.315 | 0.696 | −55.373 | |

M1255 | SVM | 0.786 | 0.450 | 0.434 | 0.548 | −280.434 | 0.7089 | 0.508 | 0.416 | 0.470 | −42.067 |

SVM+L | 0.758 | 0.424 | 0.421 | 0.574 | −290.376 | 0.730 | 1.149 | 0.626 | −0.199 | −22.467 | |

EEMD-SVM | 0.972 | 0.073 | 0.360 | 0.927 | −343.467 | 0.860 | 0.507 | 1.009 | 0.471 | 0.447 | |

STL185 | SVM | 0.931 | 0.134 | 0.488 | 0.865 | −241.064 | 0.845 | 0.283 | 0.755 | 0.704 | −13.512 |

SVM+L | 0.858 | 0.273 | 0.696 | 0.725 | −121.615 | 0.845 | 0.284 | 0.756 | 0.703 | −13.441 |

Training | Validation | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

R | NMSE | RMSE | NS | AIC | R | NMSE | RMSE | NS | AIC | ||

EEMD-ANFIS | 0.968 | 0.063 | 0.162 | 0.937 | −611.840 | 0.926 | 0.379 | 0.360 | 0.605 | −49.100 | |

M1255 | ANFIS | 0.918 | 0.157 | 0.257 | 0.842 | −457.099 | 0.785 | 0.496 | 0.412 | 0.482 | −42.602 |

ANFIS+L | 0.971 | 0.058 | 0.156 | 0.942 | −624.287 | 0.799 | 0.443 | 0.389 | 0.538 | −45.341 | |

EEMD-ANFIS | 0.982 | 0.035 | 0.249 | 0.965 | −466.512 | 0.909 | 0.183 | 0.606 | 0.809 | −24.035 | |

STL185 | ANFIS | 0.940 | 0.117 | 0.455 | 0.883 | −264.408 | 0.855 | 0.318 | 0.799 | 0.668 | −10.756 |

ANFIS+L | 0.980 | 0.039 | 0.264 | 0.961 | −447.574 | 0.910 | 0.262 | 0.726 | 0.726 | −15.360 |

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## Share and Cite

**MDPI and ACS Style**

Gong, Y.; Wang, Z.; Xu, G.; Zhang, Z.
A Comparative Study of Groundwater Level Forecasting Using Data-Driven Models Based on Ensemble Empirical Mode Decomposition. *Water* **2018**, *10*, 730.
https://doi.org/10.3390/w10060730

**AMA Style**

Gong Y, Wang Z, Xu G, Zhang Z.
A Comparative Study of Groundwater Level Forecasting Using Data-Driven Models Based on Ensemble Empirical Mode Decomposition. *Water*. 2018; 10(6):730.
https://doi.org/10.3390/w10060730

**Chicago/Turabian Style**

Gong, Yicheng, Zhongjing Wang, Guoyin Xu, and Zixiong Zhang.
2018. "A Comparative Study of Groundwater Level Forecasting Using Data-Driven Models Based on Ensemble Empirical Mode Decomposition" *Water* 10, no. 6: 730.
https://doi.org/10.3390/w10060730