# A Nonlinear Autoregressive Modeling Approach for Forecasting Groundwater Level Fluctuation in Urban Aquifers

## Abstract

**:**

^{2}) values ranging between 0.762 and 0.994 in the validation period. Comparison with other statistical models applied to the same study area shows that NARX models presented here reduced the mean absolute error (MAE) of groundwater levels forecasts by 50%. The findings of this paper are promising and provide a valuable tool for the urban city planner to assist in controlling the problem of shallow water tables for similar climatic and aquifer systems.

## 1. Introduction

## 2. Study Area and Materials

^{2}. The land use in Kuwait City is primarily residential with commercial areas. Water resources are scarce with no surface water bodies existing in the entire country. Seawater desalination is the only domestic water supply resource available. Several desalination plants were constructed to secure the increasing water demand. Due to the rapid economic development the country has witnessed during the last few decades, the population has grown rapidly to 4.5 million placing the country under a serious water shortage threat. Most of the population is concentrated within the urbanized area of Kuwait City, representing less than 2% of the total land area for the country and its suburbs near the coast.

## 3. Methodology

#### 3.1. Non-Linear Autoregressive Networks with Exogenous Input

#### 3.1.1. Network Architecture

_{y}and n

_{x}are the delay of the target and the exogenous time series respectively, and f is a nonlinear function which is generally unknown (the black box function).

#### 3.1.2. Training Algorithms

**,**$\lambda $ is a learning parameter, $I$ is the identity matrix and $e$ represents the vector of the network error.

#### 3.2. Data Preprocessing

#### 3.3. Autocorrelation and Cross-Correlation Analysis

#### 3.4. Model Validation and Performance Assessment

^{2}), the mean absolute error (MAE), and the Nash–Sutcliffe coefficient (NASH). The coefficient of determination indicates the strength of association between the observed and simulated targets, and ranges between zero and one, where zero indicates no statistical association, and one indicates a perfect match between the observed and simulated targets. R

^{2}is defined as the square of the well-known Pearson correlation coefficient. In this case, R

^{2}does not directly measure how good the predictions are (unlike how Pearson correlation does), but rather it assesses the quality of a predictor that might be constructed from the model. The following equation calculates R

^{2}for the simulated and observed groundwater depths values:

## 4. Results and Discussion

#### 4.1. Modeling Results

#### 4.2. Comparisons with Previous Studies

^{2}values obtained from the current study for all examined wells ranged between 0.762 to 0.994 which are comparable to similar studies which used similar neural network approach (e.g., Guzman et al. [14] obtained R

^{2}values ranging between 0.83 and 0.92 when using ANNs for modeling daily groundwater levels fluctuation). In addition, the performance of statistical metrics for the current study indicate that the developed modeling approach efficiency is even comparable with other data driven approaches such as support vector machine (SVM), which usually gives more accurate results [45].

## 5. Summary and Conclusions

^{2}values are increasing significantly for three of the examined wells (NZ-1A well from 0.762 to 0.966, JB-1A well from 0.887 to 0.987, and HL-1A well from 0.765 to 0.973). Comparisons with other groundwater level fluctuation modeling techniques applied to the study site showed the superiority of the NARX-based approach (MAE was reduced by 50% in comparison with periodical models). In particular, the proposed NARX model has been found to represent the variability of groundwater levels while other periodic models have failed to do so. Unlike physically based models, the proposed NARX model is simple to use and does not require detailed site investigations.

## Funding

## Conflicts of Interest

## References

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**Figure 4.**The sequential procedure for the nonlinear autoregressive modeling approach with exogenous input, (NARX)-based approach, for modeling groundwater level fluctuation in urban aquifers.

**Figure 5.**(

**A**) Groundwater levels detrending procedure for well BN-1A. (

**B**) Autocorrelation function (correlogram) applied for groundwater levels remainder at well BN-1A. Upper and lower dashed lines designate significant correlation limits. (

**C**) Cross correlation between groundwater depth remainder and monthly average temperature at well BN-1A.

**Figure 7.**Modeled versus observed groundwater levels for all examined wells with respect to the perfect 1:1 match line.

ID | Area | Easting (m) | Northing (m) | Data Period | Ground Level Above m.s.l. (m) | Water Table Depth Range (m) |
---|---|---|---|---|---|---|

BN-1A | Bayan | 503,750.6 | 243,098.7 | 1992–2001 | 27.74 | 8.42–12.23 |

NZ-1A | Nuzha | 499,402.2 | 247,349.0 | 1992–2000 | 16.45 | 2.68–4.07 |

JB-1A | Jabriya | 502,472.2 | 244,781.0 | 1993–2002 | 23.00 | 4.52–6.08 |

HL-1A | Hawally | 500,673.1 | 247,010.4 | 1993–2002 | 20.06 | 3.76–5.60 |

Auto Correlation for Remainder Data | Cross Correlation with Temperature | |||||||
---|---|---|---|---|---|---|---|---|

Well | 1st lag | 2nd lag | 3rd lag | 4th lag | 5th lag | 6th lag | Magnitude | Lag |

BN-1A | 0.92 | 0.80 | 0.65 | 0.5 | - | - | −0.48 | 3rd lag |

NZ-1A | 0.89 | 0.75 | 0.57 | - | - | - | −0.37 | 2nd lag |

JB-1A | 0.93 | 0.85 | 0.75 | 0.65 | 0.56 | 0.49 | −0.33 | 3rd lag |

HL-1A | 0.91 | 0.77 | 0.62 | 0.48 | - | - | −0.35 | 4th lag |

**Table 3.**NARX models performance at each validation round (R1: round 1, R2: round 2, and R3: round 3).

Well | Validation Round | R^{2} | MAE (m) | NASH ^{a} | ID ^{b} | FD ^{c} | HL ^{d} |
---|---|---|---|---|---|---|---|

BN-1A | R1 | 0.971 | 0.063 | 0.964 | 1:3 | 1:4 | 30 |

R2 | 0.992 | 0.026 | 0.992 | ||||

R3 | 0.994 | 0.049 | 0.993 | ||||

NZ-1A | R1 | 0.762 | 0.072 | 0.735 | 1:2 | 1:3 | 30 |

R2 | 0.967 | 0.042 | 0.966 | ||||

R3 | 0.966 | 0.048 | 0.966 | ||||

JB-1A | R1 | 0.887 | 0.020 | 0.823 | 1:3 | 1:6 | 70 |

R2 | 0.987 | 0.010 | 0.986 | ||||

R3 | 0.987 | 0.032 | 0.985 | ||||

HL-1A | R1 | 0.765 | 0.052 | 0.643 | 1:4 | 1:4 | 70 |

R2 | 0.953 | 0.050 | 0.949 | ||||

R3 | 0.973 | 0.065 | 0.964 |

^{a}NASH = Nash-Sutcliffe coefficient;

^{b}Input delays;

^{c}Feedback delays;

^{d}Hidden layers; MAE = mean absolute error, R

^{2}= coefficient of determination.

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**MDPI and ACS Style**

Alsumaiei, A.A. A Nonlinear Autoregressive Modeling Approach for Forecasting Groundwater Level Fluctuation in Urban Aquifers. *Water* **2020**, *12*, 820.
https://doi.org/10.3390/w12030820

**AMA Style**

Alsumaiei AA. A Nonlinear Autoregressive Modeling Approach for Forecasting Groundwater Level Fluctuation in Urban Aquifers. *Water*. 2020; 12(3):820.
https://doi.org/10.3390/w12030820

**Chicago/Turabian Style**

Alsumaiei, Abdullah A. 2020. "A Nonlinear Autoregressive Modeling Approach for Forecasting Groundwater Level Fluctuation in Urban Aquifers" *Water* 12, no. 3: 820.
https://doi.org/10.3390/w12030820