# Forecasting Groundwater Table in a Flood Prone Coastal City with Long Short-term Memory and Recurrent Neural Networks

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{2}of land with an average elevation of 3.2 m (above the North American Vertical Datum of 1988) and has 232 km of shoreline. Home to almost a quarter million people [47], Norfolk serves important economic and national security roles with one of the U.S.’s largest commercial ports, the world’s largest naval base, and the North American Headquarters for the North Atlantic Treaty Organization (NATO). The larger Hampton Roads Region, of which Norfolk is a major part, has the second greatest risk from sea level rise in the U.S. and is surpassed only by New Orleans [48]. This risk is partly due to coupled sea level rise and regional land subsidence from groundwater withdrawals from the deep Potomac Aquifer for water supply and glacial isostatic adjustment [49]. Because of these and other factors, including low relief terrain and a regular hurricane season, the city and larger Hampton Roads region face increasingly frequent and severe recurrent flooding [4] which threatens its economic, military, and historic importance.

#### 2.2. Data

#### 2.2.1. Observed Data

#### 2.2.2. Forecast Data

^{2}. These data are archived by the Center for High Performance Computing at the University of Utah [53] and was accessed from that database.

#### 2.3. Methodology

#### 2.3.1. Input Data Preprocessing

**gwl**, rainfall

_{I}**rain**, and sea level

**sea**. Each row in the label tensor is a vector of groundwater table values

**gwl**to be predicted (Table 3).

_{L}_{fcst}that includes both observed and forecast data as specified in Figure 3. The same normalization from 0–1 used for the observed data was applied to the forecast data.

#### 2.3.2. Input Variable Cross-Correlation Analysis

_{R}between rainfall and groundwater table response and δ

_{S}between sea level and groundwater table response. The appropriate δ

_{R}and δ

_{S}, in hours, for each well was approximated by a cross correlation analysis [25]. This process involves shifting one signal in relation to the other until a rainfall or sea level observation lines up with its corresponding groundwater table response. The highest cross correlation value (CCF) corresponds to the most influential δ

_{R}or δ

_{S}.

#### 2.3.3. Storm Event Response Identification

_{full}represents the continuous time series data and includes both dry and wet days. The second training set D

_{storm}consists only of time periods that were identified as storm events. D

_{storm}was created through a filtering process using the gradient and peaks of the observed groundwater table values. For any storm event, the starting time of the event was based on locating the local maxima of the gradient of the groundwater table and looking backward in time to the first occurrence of zero gradient. A peak finding algorithm [58] was then used to locate the peak of the groundwater table that occurred after the corresponding starting time; peak values were used as the end point of the storm.

#### 2.3.4. Bootstrapping Datasets

_{full}datasets in a manner appropriate for time series data, circular block bootstrapping with replacement was used [59]. The block size was based on the average storm length found when creating the storm datasets for each well. Because the D

_{storm}datasets were already separated into blocks of different time periods, they were bootstrapped by randomly sampling the blocks with replacement. By creating one thousand bootstrap replicates of each dataset, a normal distribution of error can be approximated when the models are trained and tested. The first 70% of each bootstrapped dataset was taken as the training data and the remaining 30% was used as the test set.

#### 2.3.5. Recurrent Neural Networks

_{t}is the hidden state, y

_{t}is the output, and x

_{t}is the input vector. The input, hidden, and output weights are represented by W, U, and V, respectively, and b is the bias. The hyperbolic tangent activation function is noted as tanh.

#### 2.3.6. Long Short-term Memory Neural Networks

_{t}, i

_{t}, and o

_{t}represent the forget, input, and output gates, respectively. The new cell state candidate values and updated cell state are represented by C′

_{t}and C

_{t}, respectively. Element-wise multiplication of vectors is represented by ° and the sigmoid activation function is noted as σ.

#### 2.3.7. Hyperparameter Tuning

#### 2.3.8. Model Training and Evaluation

_{full}and the D

_{storm}datasets (Figure 5). At each time step, models were fed input data and output a vector of forecast groundwater table levels, as shown in Table 3. During training, the models sought to minimize the cost function, which is the RMSE between predicted and observed values, by iteratively adjusting the network weights. After training, the D

_{full}models were tested on the D

_{full}, D

_{storm}, and D

_{fcst}test sets. Likewise, the D

_{storm}models were tested on the D

_{storm}and D

_{fcst}test sets.

#### 2.3.9. Results Post-Processing

_{full}dataset (Table 5, Comparison ID A). The hypotheses were evaluated using t-tests to evaluate whether or not there was a statistically significant difference between the mean of the 1000 RMSEs between two models [70]. In order to reject a null hypothesis that the two models have identical average values, the p-value from the t-test would need to be significant (less than 0.01).

## 3. Results

#### 3.1. Data Preprocessing Results

#### 3.1.1. Input Variable Cross-Correlation Analysis

_{R}were generally expected to increase with a greater distance between the land surface and the groundwater table. It was found δ

_{R}did increase with greater depth to the groundwater table when GW2 and GW3 were compared. At GW2, δ

_{R}was 26 h and the mean groundwater table depth was 0.61 m (Table 1) while at GW3 δ

_{R}was 59 h and the mean groundwater table depth was 2.32 m. At the other wells, however, this relationship did not hold. For example, GW1 had the same δ

_{R}as GW2, but the mean groundwater table depth was very similar to that of GW3 (2.31 m). Other characteristics that influence infiltration rate, such as vertical hydraulic conductivity, porosity, impermeable surfaces, or the configuration of the stormwater system appear to have had a large effect on δ

_{R}at these wells. In addition, sea level may also be influencing groundwater table levels at some or all of these wells.

_{S}on the groundwater table was more difficult to determine than rainfall lags δ

_{R}, indicating that sea level does not have as much impact on certain wells; there did not seem to be clear correlations for GW3, GW5, or GW6. It was expected that the impact of sea level would decrease with greater distance between a given well and the closest tidal waterbody influencing it. However, this did not seem to have a strong relationship. GW4, for example, was the farthest well from a tidal water body but had the shortest δ

_{S}, suggesting that tidal water may have a more direct route to this location. While a strong correlation between sea level and groundwater table was not found for three wells, it was deemed that sea level could still be an important input variable for models at those wells because of their proximity to tidal water bodies [71,72]. In order to keep the data preprocessing consistent, and because δ

_{S}values could not be found for all wells and the δ

_{S}values found were always shorter than δ

_{R}values, δ

_{R}was taken as the lag value for all input variables.

#### 3.1.2. Storm Event Response Identification

_{full}datasets. The storm events identified for each well also accounted for the majority of total rainfall, indicating that the method is capturing large rainfall events. Storm surge is also being captured at most wells as shown by the positive increase in mean sea level for the storm events compared to the D

_{full}datasets (Table 7). Figure 6 shows an example of storms found with this process; large responses of the groundwater table are captured, but smaller responses are excluded.

#### 3.1.3. Hyperparameter Tuning

^{−3}[73]. The largest number of neurons possible (75) was used in five of the seven RNN (Table 8) and LSTM (Table 9) models. The other models of each type used a midrange number of neurons (40 or 50).

#### 3.2. Model Results

#### 3.2.1. Network and Training Data Type Comparison

_{full}or D

_{storm}, LSTM models have lower mean RMSE values than RNN models (Figure 7A,B), as hypothesized (Table 5, A and B). LSTM models trained and tested with D

_{full}had average RMSE values that were lower than RNN models by 49%, 38%, and 18% for the t + 1, t + 9, and t + 18 predictions, respectively. LSTM’s advantage over RNN decreased as the prediction horizon increased. Similarly, LSTM models trained and tested with D

_{storm}had lower average RMSE values than RNN models by 50%, 55%, and 36% for the t + 1, t + 9, and t + 18 predictions when tested on D

_{storm}, respectively.

_{storm}, the models trained with D

_{storm}outperformed the models trained with D

_{full}(Figure 7C,D), with the exception of the RNN for GW4. In this scenario, the models trained with D

_{storm}had RMSE values that were lower than models trained with D

_{full}by an average of 33%, 39%, and 42% for the RNN models and by an average of 40%, 58%, and 56% for the LSTM models for the t + 1, t + 9, and t + 18 predictions, respectively. The improvement in performance when using D

_{storm}as opposed to D

_{full}, increased with the prediction horizon. While this was true for both model types, the performance improvement for LSTM was greater than for the RNN.

_{storm}(Figure 7B,C) had a larger RMSE for the t + 9 prediction than the t + 18 prediction. This pattern is the same for the GW6 RNN (Figure 7A,C) and may have been caused by some combination of hyperparameters and/or some unknown error in the dataset. Causes of individual errors in these types of models, however, are very difficult to pinpoint [25].

#### 3.2.2. Real-Time Forecast Scenario

_{fcst}test set are shown in Figure 8 and correspond to hypotheses E–H (Table 5). The distributions of RMSE values for all bootstrap models in this subsection is available in Appendix B; corresponding MAE values are available in Appendix C.

_{storm}(Figure 8F–H) performed much better than those trained with D

_{full}(Figure 8E), which had RMSE values of up to nearly 1.25 m. In contrast to the difference training data type made, model architecture only made a small difference in performance (Figure 8E,F). All differences seen in Figure 8E were statistically significant at the 0.001 level, except GW3 at t + 9 and GW6 at t + 1 where the results were almost identical. The comparisons in Figure 8F–H all had significant p-values.

_{fcst}as input data. The forecasts at GW1 are shown in Figure 9 for Tropical Storm Julia, which impacted Norfolk in late September of 2016. The initial rainfall from this storm on the 19th caused the groundwater table to spike early on the 20th. Subsequent rainfall on the 20th, 21st, and 22nd maintained the elevated groundwater table level. The LSTM model trained with D

_{full}has greatly increasing error as the forecast horizon grows (Figure 9 t + 1, t + 9, t + 18) and tends to be overly impacted by sea level fluctuations. In contrast, the predicted groundwater table level from the LSTM model trained with D

_{storm}has much better agreement with the observed groundwater table levels, even as the forecast horizon increases.

## 4. Discussion

_{storm}outperformed models trained with the full dataset D

_{full}when tested on either observed for forecast data. In the real-time scenario one reason for this difference in performance could be the structure of the test set D

_{fcst}. These results indicate that the structure of the time series data in D

_{storm}and D

_{fcst}are more closely aligned, as opposed to the time series structure of D

_{full}and D

_{fcst}. The models trained on D

_{full}also have to learn groundwater table response with many observations where no rainfall occurred. In contrast, models trained on D

_{storm}, which have a higher proportion of observations with rainfall, may have a clearer pattern to learn.

_{storm}demonstrated predictive skill, forecasting groundwater table levels with low RMSE values (Figure 8F). Models trained with D

_{full}however performed much worse because of the noisier signal they had to learn (Figure 9) and are not suitable for use in a real-time forecasting scenario. Across all wells, averaged RMSE values for the RNN models were 0.06 m, 0.1 m, and 0.1 m for the t + 1, t + 9, and t + 18 predictions, respectively. Averaged RMSE values for the LSTMs were slightly lower at 0.03 m, 0.05 m, and 0.07 m for the t+1, t+9, and t+18 predictions, respectively. While there is limited research on the use of LSTMs for forecasting groundwater table, these results are comparable with the work of J. Zhang et al. [46], who reported RMSE values for one-step ahead prediction of monthly groundwater table at six sites ranging from 0.07 m to 0.18 m. The current work makes advances by showing that both LSTM and RNN can accurately forecast groundwater table response to storm events at an hourly time step, with forecast input data, and at longer prediction horizons all of which are necessary in a coastal urban environment.

_{full}and D

_{storm}data sets and retraining and retesting the models. Of the wells that were not correlated with sea level, GW3 and GW6 performed better without sea level data. Using RNN models trained with D

_{full}, there was an average decrease in RMSE of 12% for GW3 and 41% for GW6. The only exception to this is the GW6 RNN trained with D

_{storm}which performed much worse without sea level. For LSTM models trained with D

_{full}however, there was only a 3% decrease in RMSE for GW3 and a 2% decrease for GW6. The third well that was not correlated with sea level, GW5, was worse without sea level for the RNN trained with D

_{full}; the average increase in RMSE was 17%. Removing sea level at this well had no change in RMSE for the LSTM models trained with D

_{full}. This particular well is only 32 m from the coast so the influence of sea level seems reasonable. When models were trained with D

_{storm}excluding sea level, across all well there was an average increase in RMSE of 8% for RNN models and no change for LSTM models. This demonstrates that sea level data is important for groundwater table prediction during storms for wells close to the coast and this is captured effectively by the D

_{storm}datasets (Table 7). This analysis indicated that RNN models were much more sensitive to the inputs used than LSTM models. As designed, the structure of LSTM models allowed them to filter out noisy data and have little to no change in RMSE if sea level data was removed, especially when using the best performing combination of LSTM and D

_{storm}training data.

_{storm}in the real-time forecasting scenario were statistically significant (Figure 8F). The average difference in the RNN and LSTM RMSE values, however, was only 0.03 m, 0.05 m, and 0.03 m for the t + 1, t + 9, and t + 18 predictions, respectively. If these groundwater table forecasts were to be used as additional input to a rainfall-runoff model to predict flooding, it seems unlikely that the small differences between RNN and LSTM models would have a large impact, especially when compared to other factors like rainfall variability and storm surge timing.

## 5. Conclusions

_{full}and a dataset of only time periods with storm events D

_{storm}, were bootstrapped and used to train and test the models. An additional dataset D

_{fcst}including forecasts of rainfall and sea level was used to evaluate model performance in a simulation of real-time model application. Statistical significance in model performance was evaluated with t-tests.

- Both model type and training data are important factors in creating skilled predictions of hourly groundwater table using observed data:
- Using D
_{full}, LSTM had a lower average RMSE than RNN (0.09 m versus 0.14 m, respectively) - Using D
_{storm}, LSTM had a lower average RMSE than RNN (0.05 m versus 0.10 m, respectively)

- The best predictive skill was achieved using LSTM models trained with D
_{storm}(average RMSE = 0.05 m) versus RNN models trained with D_{storm}(average RMSE = 0.10 m) - LSTM has better performance than RNN but requires approximately 3 times more time to train
- In a real-time scenario using observed and forecasted input data, accurate forecasts of groundwater table were created with an 18 h horizon:
- LSTM: Average RMSE values of 0.03, 0.05, and 0.07 m, for the t + 1, t + 9, and t + 18h forecasts, respectively
- RNN: Average RMSE values of 0.06, 0.10, and 0.10 m, for the t + 1, t + 9, and t + 18h forecasts, respectively

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**RMSE distributions for GW1 using observed data. Columns represent the forecast horizons t + 1, t + 9, and t + 18. Rows are specified as model type, training data, and testing data.

**Figure A2.**RMSE distributions for GW2 using observed data. Columns represent the forecast horizons t + 1, t + 9, and t + 18. Rows are specified as model type, training data, and testing data.

**Figure A3.**RMSE distributions for GW3 using observed data. Columns represent the forecast horizons t + 1, t + 9, and t + 18. Rows are specified as model type, training data, and testing data.

**Figure A4.**RMSE distributions for GW4 using observed data. Columns represent the forecast horizons t + 1, t + 9, and t + 18. Rows are specified as model type, training data, and testing data.

**Figure A5.**RMSE distributions for GW5 using observed data. Columns represent the forecast horizons t + 1, t + 9, and t + 18. Rows are specified as model type, training data, and testing data.

**Figure A6.**RMSE distributions for GW6 using observed data. Columns represent the forecast horizons t + 1, t + 9, and t + 18. Rows are specified as model type, training data, and testing data.

**Figure A7.**RMSE distributions for GW7 using observed data. Columns represent the forecast horizons t + 1, t + 9, and t + 18. Rows are specified as model type, training data, and testing data.

## Appendix B

**Figure A8.**RMSE distributions for GW1 using forecast input data. Columns represent the forecast horizons t + 1, t + 9, and t + 18. Rows are specified as model type, training data, and testing data.

**Figure A9.**RMSE distributions for GW2 using forecast input data. Columns represent the forecast horizons t + 1, t + 9, and t + 18. Rows are specified as model type, training data, and testing data.

**Figure A10.**RMSE distributions for GW3 using forecast input data. Columns represent the forecast horizons t + 1, t + 9, and t + 18. Rows are specified as model type, training data, and testing data.

**Figure A11.**RMSE distributions for GW4 using forecast input data. Columns represent the forecast horizons t + 1, t + 9, and t + 18. Rows are specified as model type, training data, and testing data.

**Figure A12.**RMSE distributions for GW5 using forecast input data. Columns represent the forecast horizons t + 1, t + 9, and t + 18. Rows are specified as model type, training data, and testing data.

**Figure A13.**RMSE distributions for GW6 using forecast input data. Columns represent the forecast horizons t + 1, t + 9, and t + 18. Rows are specified as model type, training data, and testing data.

**Figure A14.**RMSE distributions for GW7 using forecast input data. Columns represent the forecast horizons t + 1, t + 9, and t + 18. Rows are specified as model type, training data, and testing data.

## Appendix C

**Table A1.**Mean mean absolute error (MAE) values for each model type and training dataset treatment at each well and forecast period when tested on observed data.

Model Type | Training Data | Testing Data | Forecast Period | GW1 | GW2 | GW3 | GW4 | GW5 | GW6 | GW7 |

RNN | D_{full} | D_{full} | t + 1 | 0.031 | 0.060 | 0.072 | 0.019 | 0.035 | 0.116 | 0.029 |

t + 9 | 0.049 | 0.089 | 0.099 | 0.036 | 0.054 | 0.410 | 0.044 | |||

t + 18 | 0.069 | 0.118 | 0.127 | 0.052 | 0.075 | 0.236 | 0.060 | |||

RNN | D_{full} | D_{storm} | t + 1 | 0.038 | 0.072 | 0.080 | 0.022 | 0.047 | 0.121 | 0.034 |

t + 9 | 0.064 | 0.114 | 0.119 | 0.042 | 0.074 | 0.397 | 0.054 | |||

t + 18 | 0.092 | 0.151 | 0.157 | 0.060 | 0.102 | 0.228 | 0.076 | |||

LSTM | D_{full} | D_{full} | t + 1 | 0.020 | 0.029 | 0.021 | 0.008 | 0.016 | 0.013 | 0.014 |

t + 9 | 0.040 | 0.067 | 0.053 | 0.027 | 0.039 | 0.032 | 0.028 | |||

t + 18 | 0.061 | 0.102 | 0.087 | 0.046 | 0.063 | 0.052 | 0.045 | |||

LSTM | D_{full} | D_{storm} | t + 1 | 0.025 | 0.033 | 0.026 | 0.010 | 0.020 | 0.013 | 0.016 |

t + 9 | 0.056 | 0.083 | 0.070 | 0.032 | 0.053 | 0.034 | 0.036 | |||

t + 18 | 0.084 | 0.128 | 0.116 | 0.054 | 0.084 | 0.057 | 0.058 | |||

RNN | D_{storm} | D_{storm} | t + 1 | 0.030 | 0.060 | 0.069 | 0.069 | 0.039 | 0.026 | 0.026 |

t + 9 | 0.041 | 0.068 | 0.080 | 0.288 | 0.045 | 0.028 | 0.033 | |||

t + 18 | 0.051 | 0.085 | 0.095 | 0.208 | 0.048 | 0.036 | 0.043 | |||

LSTM | D_{storm} | D_{storm} | t + 1 | 0.024 | 0.031 | 0.023 | 0.008 | 0.017 | 0.012 | 0.013 |

t + 9 | 0.036 | 0.049 | 0.037 | 0.015 | 0.027 | 0.019 | 0.021 | |||

t + 18 | 0.045 | 0.066 | 0.052 | 0.023 | 0.033 | 0.027 | 0.029 |

**Table A2.**Mean MAE values for each model type and training dataset treatment at each well and forecast period when tested on forecast data D

_{fcst}.

Model Type | Training Data | Testing Data | Forecast Period | GW1 | GW2 | GW3 | GW4 | GW5 | GW6 | GW7 |

RNN | D_{full} | D_{fcst} | t + 1 | 0.211 | 0.308 | 0.881 | 0.206 | 0.613 | 0.369 | 0.356 |

t + 9 | 0.439 | 0.513 | 1.001 | 0.333 | 0.668 | 0.960 | 0.608 | |||

t + 18 | 0.998 | 0.537 | 1.131 | 0.800 | 0.913 | 0.493 | 1.113 | |||

LSTM | D_{full} | D_{fcst} | t + 1 | 0.235 | 0.454 | 0.716 | 0.199 | 0.394 | 0.346 | 0.295 |

t + 9 | 0.374 | 0.362 | 0.976 | 0.285 | 0.759 | 0.440 | 0.853 | |||

t + 18 | 0.939 | 0.421 | 1.178 | 0.764 | 1.011 | 0.488 | 1.222 | |||

RNN | D_{storm} | D_{fcst} | t + 1 | 0.027 | 0.064 | 0.064 | 0.068 | 0.027 | 0.026 | 0.023 |

t + 9 | 0.032 | 0.060 | 0.096 | 0.241 | 0.037 | 0.026 | 0.036 | |||

t + 18 | 0.038 | 0.073 | 0.106 | 0.160 | 0.034 | 0.037 | 0.034 | |||

LSTM | D_{storm} | D_{fcst} | t + 1 | 0.022 | 0.029 | 0.027 | 0.007 | 0.014 | 0.012 | 0.012 |

t + 9 | 0.028 | 0.044 | 0.038 | 0.012 | 0.019 | 0.019 | 0.017 | |||

t + 18 | 0.037 | 0.059 | 0.055 | 0.022 | 0.025 | 0.027 | 0.025 |

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**Figure 2.**Hourly groundwater table level, sea level, and rainfall at individual wells for Tropical Storm Julia.

**Figure 3.**Study workflow detailing major steps in the data preprocessing, neural network modeling, and results post-processing.

**Figure 4.**Recurrent neural network (RNN) (

**A**) and long short-term memory (LSTM) (

**B**) model architectures. Merging lines show concatenation and splitting lines represent copies of matrices being sent to different locations.

**Figure 7.**Mean root mean squared error (RMSE) values for each model type and training dataset treatment at each well and forecast period. Subplot letters correspond to the hypothesis being tested (Table 5) and are comparisons of (

**A**) RNN and LSTM models trained and tested with D

_{full}(

**B**) RNN and LSTM models trained and tested with D

_{storm}(

**C**) RNN models trained with either D

_{full}or D

_{storm}and tested on D

_{storm}(

**D**) LSTM models trained with either D

_{full}or D

_{storm}and tested on D

_{storm}.

**Figure 8.**Mean RMSE values from the forecast test set D

_{fcst}for each model type and training dataset treatment at each well and forecast period. Subplot letters correspond to the hypothesis being tested (Table 5) and are comparisons of (

**E**) RNN and LSTM models trained with D

_{full}(

**F**) RNN and LSTM models trained with D

_{storm}(

**G**) RNN models trained with either D

_{full}or D

_{storm}(

**H**) LSTM models trained with either D

_{full}or D

_{storm}.

**Figure 9.**Comparison of groundwater table observations and forecasts at GW1 from LSTM models trained with the D

_{full}and D

_{storm}training sets.

Well ID | Land Surface Elevation (m) ^{a} | Well Depth (m) ^{b} | Distance to Tidal Water (m) | Impervious Area (%) ^{c} | Groundwater Table Level (m) ^{a,d} | ||
---|---|---|---|---|---|---|---|

Minimum | Maximum | Mean | |||||

GW1 | 2.21 | 4.27 | 36 | 27 | −0.678 | 0.883 | −0.102 |

GW2 | 1.24 | 4.08 | 32 | 23 | −0.670 | 1.476 | 0.635 |

GW3 | 4.35 | 5.18 | 668 | 42 | 1.197 | 3.844 | 2.026 |

GW4 | 3.24 | 4.57 | 777 | 53 | 0.659 | 2.021 | 1.075 |

GW5 | 1.72 | 2.53 | 32 | 20 | −0.167 | 1.5562 | 0.492 |

GW6 | 2.35 | 3.23 | 41 | 30 | 0.259 | 2.012 | 0.742 |

GW7 | 2.57 | 4.60 | 650 | 73 | 0.200 | 1.750 | 0.707 |

^{a}Referenced to North American Vertical Datum of 1988 (NAVD88);

^{b}Below land surface;

^{c}Percent of area classified as impervious within a 610 m buffer around well;

^{d}Statistics calculated from January 2010 to May 2018.

Well ID | Rain Gauge (s) |
---|---|

GW1 | R1 |

R2 | |

R4 | |

R7 | |

GW2 | R4 |

GW3 | R2 |

GW4 | R1 |

R3 | |

R5 | |

R7 | |

GW5 | R2 |

R6 | |

GW6 | R7 |

GW7 | R6 |

Inputs | Labels |
---|---|

gwl = {t − δ…t}_{I} | gwl = {t + 1…t + τ }_{L} |

rain = {t − δ…t + τ } | |

sea = {t − δ…t + τ } |

Hyperparameter | Type | Options Explored |
---|---|---|

Number of Neurons | Choice | 10, 15, 20, 40, 50, 75 |

Activation Function | Choice | Rectified Linear Unit (relu), Hyperbolic tangent (tanh), Sigmoid |

Optimization Function | Choice | Adam, Stochastic Gradient Descent (SGD), Root Mean Square Propagation (RMSProp) |

Learning Rate | Choice | 1 × 10^{−3}, 1 × 10^{−2}, 1 × 10^{−1} |

Dropout Rate | Continuous | 0.1–0.5 |

Comparison ID | Null Hypothesis | Testing Data |
---|---|---|

A | RMSE(LSTM, D_{full}) = RMSE(RNN, D_{full}) | D_{full} |

B | RMSE(LSTM, D_{storm}) = RMSE(RNN, D_{storm}) | D_{storm} |

C | RMSE(RNN, D_{storm}) = RMSE(RNN, D_{full}) | |

D | RMSE(LSTM, D_{storm}) = RMSE(LSTM, D_{full}) | |

E | RMSE(LSTM, D_{full}) = RMSE(RNN, D_{full}) | D_{fcst} |

F | RMSE(LSTM, D_{storm}) = RMSE(RNN, D_{storm}) | |

G | RMSE(RNN, D_{storm}) = RMSE(RNN, D_{full}) | |

H | RMSE(LSTM, D_{storm}) = RMSE(LSTM, D_{full}) |

Well ID | δ_{R} (h) | δ_{S} (h) |
---|---|---|

GW1 | 26 | 19 |

GW2 | 26 | 18 |

GW3 | 59 | -- |

GW4 | 25 | 17 |

GW5 | 28 | -- |

GW6 | 48 | -- |

GW7 | 58 | 51 |

Well ID | Average Storm Duration (h) | Number of Events | % of Total Rain | % Increase in Mean Sea Level over D_{full} |
---|---|---|---|---|

GW1 | 83 | 239 | 75 | 27 |

GW2 | 82 | 307 | 85 | 36 |

GW3 | 137 | 155 | 57 | 18 |

GW4 | 89 | 254 | 67 | 18 |

GW5 | 91 | 149 | 60 | 64 |

GW6 | 120 | 295 | 60 | 0 |

GW7 | 132 | 166 | 63 | 0 |

Well | Dropout Rate | Activation Function | Optimization Function | Learning Rate | Neurons |
---|---|---|---|---|---|

GW1 | 0.126 | tanh | adam | 10^{−3} | 40 |

GW2 | 0.340 | tanh | adam | 10^{−3} | 75 |

GW3 | 0.320 | tanh | adam | 10^{−3} | 75 |

GW4 | 0.111 | tanh | adam | 10^{−3} | 75 |

GW5 | 0.127 | relu | adam | 10^{−3} | 75 |

GW6 | 0.145 | tanh | adam | 10^{−3} | 75 |

GW7 | 0.104 | tanh | adam | 10^{−3} | 40 |

Well | Dropout Rate | Activation Function | Optimization Function | Learning Rate | Neurons |
---|---|---|---|---|---|

GW1 | 0.355 | tanh | adam | 10^{−3} | 75 |

GW2 | 0.106 | tanh | adam | 10^{−3} | 40 |

GW3 | 0.166 | tanh | adam | 10^{−3} | 75 |

GW4 | 0.102 | tanh | adam | 10^{−3} | 75 |

GW5 | 0.103 | tanh | adam | 10^{−3} | 50 |

GW6 | 0.251 | tanh | adam | 10^{−3} | 75 |

GW7 | 0.177 | tanh | adam | 10^{−3} | 75 |

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## Share and Cite

**MDPI and ACS Style**

Bowes, B.D.; Sadler, J.M.; Morsy, M.M.; Behl, M.; Goodall, J.L. Forecasting Groundwater Table in a Flood Prone Coastal City with Long Short-term Memory and Recurrent Neural Networks. *Water* **2019**, *11*, 1098.
https://doi.org/10.3390/w11051098

**AMA Style**

Bowes BD, Sadler JM, Morsy MM, Behl M, Goodall JL. Forecasting Groundwater Table in a Flood Prone Coastal City with Long Short-term Memory and Recurrent Neural Networks. *Water*. 2019; 11(5):1098.
https://doi.org/10.3390/w11051098

**Chicago/Turabian Style**

Bowes, Benjamin D., Jeffrey M. Sadler, Mohamed M. Morsy, Madhur Behl, and Jonathan L. Goodall. 2019. "Forecasting Groundwater Table in a Flood Prone Coastal City with Long Short-term Memory and Recurrent Neural Networks" *Water* 11, no. 5: 1098.
https://doi.org/10.3390/w11051098