Daily Groundwater Level Prediction and Uncertainty Using LSTM Coupled with PMI and Bootstrap Incorporating Teleconnection Patterns Information

Abstract

Daily groundwater level is an indicator of groundwater resources. Accurate and reliable groundwater level (GWL) prediction is crucial for groundwater resources management and land subsidence risk assessment. In this study, a representative deep learning model, long short-term memory (LSTM), is adopted to predict groundwater level with the selected predictors by partial mutual information (PMI), and bootstrap is employed to generate different samples combination for training many LSTM models, and the predicted values by many LSTM models are used for the uncertainty assessment of groundwater level prediction. Two wells of different climate zones in the USA were used as a case study. Different significant predictors of GWL for two wells were identified by PMI from candidate predictors incorporating teleconnection patterns information. The results show that GWL is significantly affected by antecedent GWL, AO, Niño 3.4, Niño 1 + 2, and precipitation in humid areas, and by antecedent GWL, AO, Niño 3.4, Niño 3, Niño 1 + 2, and PNA in arid areas. Predictor selection can assist in improving the prediction performance of the LSTM model. The relationship between GWL and significant predictors were modeled by the LSTM model, and it achieved higher accuracy in humid areas, while the performance in arid areas was poorer due to limited precipitation information. The performance of LSTM was improved by increasing correlation coefficient (R²) values by 10% and 25% for 2 wells compared to generalized regression neural network (GRNN). Three uncertainty evaluation metrics indicate that LSTM reduced the uncertainty compared to GRNN model. LSTM coupling with PMI and bootstrap can be a promising approach for accurate and reliable groundwater level prediction for different climate zones.

1. Introduction

Groundwater level is an indicator of groundwater resources. The rise and fall of groundwater level can reflect the input and output of atmosphere, surface water, and underlying surface system to groundwater system [1,2]. In some areas, the groundwater level is only affected by climate change; for example, the increase of rainfall can raise the groundwater level and drought can lower the groundwater level [3]. In addition to climate change, the groundwater level of some areas is also affected by human activities. For example, artificial recharge for ecological restoration can raise the groundwater level, and artificial mining can lower the groundwater level.

Accurate and reliable daily groundwater level (GWL) prediction is crucial for groundwater resources management and land subsidence risk assessment [4,5,6]. Artificial intelligence (AI) has attracted a lot of attention as it can model the relationship between groundwater level and various influencing variables through extracting groundwater dynamic characteristics from large amounts of observation data [7,8,9]. Predictor selection, model structure, parameters, and uncertainty are the main concerns in the AI-based GWL prediction process [10,11,12].

There are large amounts of candidate predictors for GWL prediction including rainfall, historical GWL, teleconnection patterns, and so on [13,14]. Holman et al. [15] investigated non-stationary groundwater level response to North Atlantic Ocean–atmosphere teleconnection patterns, and the results indicated that there was common statistically significant correlation between the teleconnection indices and groundwater levels. Lee et al. [16] predicted groundwater levels using an ANN model with input variables composed of river level and two anthropogenic factors in Yangpyeong riverside area, South Korea. Salam et al. [17] explored the relationship between groundwater level and El-Niño Southern Oscillation (ENSO) teleconnection indices and predicted the GWL changes by autoregressive integrated moving average (ARIMA) modeling. Yin et al. [18] employed precipitation, temperature, and terrestrial water storage as input variables to predict monthly GWL. Kajewska-Szkudlarek et al. [19] provided the accurate GWL prediction considering monthly temperature and precipitation as well as historical GWL time series. Partial mutual information (PMI) can be effective for selecting the significant predictors because of avoiding the feature redundancy without assuming any dependence structure among the candidate predictors [20,21]. It is a necessary step for developing GWL prediction model to identify the significant predictors in different climate zones [22].

Various statistical algorithms and AI have been used for groundwater level prediction [23,24,25,26]. Nourani et al. [27] evaluated the feasibility of artificial neural network (ANN) methodology for estimating the groundwater levels in an aquifer in northwestern Iran. Mirzavand and Ghazavi [28] applied time series methods such as seasonal auto-regressive integrated moving average (SARIMA) for groundwater level forecasting in an arid environment. Rahman et al. [29] forecasted average monthly GWL at 1, 2, and 3 months ahead for seven wells in Kumamoto City, Japan, using eXtreme Gradient Boosting and Random Forests coupling with wavelet transforms. Azari et al. [30] proposed linear stochastic approaches integrated with preprocessing techniques for groundwater level forecasting and compared with artificial neural network and adaptive neuro-fuzzy inference systems. Ghasemlounia et al. [31] developed a novel framework for forecasting groundwater level fluctuations using bi-directional long short-term memory (BiLSTM) deep neural network. Wunsch et al. [32] employed ANN, LSTM, convolutional neural networks (CNNs), and non-linear autoregressive networks with exogenous input (NARX) for groundwater level forecasting. Recently, LSTM model has been widely applied to predict groundwater levels which can overcome the vanishing and exploding gradient problems of the classical RNNs. However, the applicability of LSTM model should be studied further in different climate zones.

Different sources of uncertainty include model inputs, parameters, and structure for AI-based prediction models. Kasiviswanathan et al. [33] employed genetic programming to predict monthly groundwater level with model uncertainty quantification. Takafuji et al. [34] combined ARIMA models and sequential Gaussian simulation (SGS) for groundwater level prediction and uncertainty in the Bauru aquifer, Brazil. Nourani et al. [35] integrated artificial neural network (ANN) and lower upper bound estimation (LUBE) method for uncertainty assessment of groundwater level predictions in the Ardebil plain, Iran. Uncertainty approach be divided into two categories: (1) the model can directly give the probability prediction results and the model needs to integrate Monte Carlo, Bayesian model averaging (BMA), or bootstrap for uncertainty quantification [36]. Model inputs have a great influence on model prediction because model parameters and structure can also be influenced by model inputs. Bootstrap as a resampling approach can be a more effective approach for estimating model input uncertainty [37,38].

This study proposes LSTM coupled with PMI and bootstrap for daily groundwater level prediction and uncertainty and test the applicability with two wells of different climate zones in Arizona and Iowa, USA. The objectives of the paper are as follows: (1) to identify significant predictors for GWLs of different climate zones, respectively, (2) to determine the optimal LSTM model structure and parameter values and compare the prediction results with GRNN model, and (3) to evaluate the prediction uncertainty using LSTM integrated with bootstrap.

2. Materials and Methods

2.1. Case Study

Two wells in the United States were selected for research. A map of study area and the location of 2 wells is shown in the Figure 1. Well 1 (USGS code: 430510093425101) is located in Hancock County in Iowa and has a length time from 4 July 2008 to 8 April 2022. Iowa has a temperate continental climate: cold in winter and hot in summer. The average temperature in January is −9 °C in the northwest and −4 °C in the southeast, which can drop to −34 °C in case of severe storm. The average daytime temperature in July is 34 °C, which is very hot. Tropical ocean air masses from the Gulf of Mexico often bring thunderstorms. The average annual precipitation in the northwest is 711 mm and 864 mm in the south. Most of them fall in summer. There is less snow in winter than in the eastern and northern states. Drought often occurs, especially in the northwest. Well 2 (USGS code: 364338110154601) is located in Navajo County in Arizona and has a length time from 8 June 2005 to 11 April 2022. Arizona is located in the southwest of the United States, with Phoenix as its capital, covering an area of 294,000 km². It is the sixth largest state in the United States. The climate of the whole prefecture changes greatly. The average temperature in Flagstaff, located in the north of the central part, is −3 °C in January and 19 °C in July. The average temperature in Phoenix, the capital, is 11 °C in January and 33 °C in July. The natural conditions in the prefecture are bad and the surface water resources are very scarce. It belongs to a dry desert climate. Although there are many rivers in Arizona that provide it with rich surface water resources, water is still very tight due to the wide distribution of high-temperature and arid deserts in the state. For ordinary years, the state’s annual evaporation in July can reach 18.5 billion m³. Since the 1960s, Arizona’s annual water demand has risen sharply, the supply of surface water is insufficient, and a large amount of groundwater has to be exploited. Before the central Arizona Project was completed, groundwater accounted for more than 60% of the total water supply. In the early 1960s, the total annual water demand was 5.3 billion m³, the surface water provided by rivers was only 1.6 billion m³, and the remaining 3.7 billion m³ came from groundwater. The annual available effective precipitation infiltration recharge of groundwater in Arizona is only 860 million m³, and the annual overexploitation of groundwater reaches 2.84 billion m³. Due to the overexploitation of groundwater, the groundwater level in the state continued to decline. According to the statistics of Arizona water resources department and the observation, investigation, and statistics of the groundwater level, the average annual decline of groundwater level is about 0.9–1.8 m.

In this study, six statistical indexes were used to describe the general characteristics of the groundwater levels of two wells in Arizona and Iowa, USA: maximum, minimum, mean, variance, skewness, and kurtosis, as shown in

Table 1. Statistics of observed groundwater levels for two wells in Arizona and Iowa, USA.

Characteristics	Well 1	Well 2
location	Hancock County, Iowa	Navajo County, Arizona
USGS code	USGS 430510093425101	USGS 364338110154601
latitude	43°05′10″	34°43′38″
longitude	93°42′51″	110°15′45″
maximum (m)	2.02	54.12
minimum (m)	0.05	46.15
variance	0.15	2.42
mean (m)	0.99	50.11
kurtosis	3.17	2.54
skewness	0.57	0.03

. The maximum value of well 1 was 2.02 m, and the minimum value was 0.05 m, with a difference of 1.97 m. The maximum value and minimum value of well 2 are 54.12 m and 46.15 m, respectively, with a difference of 7.97 m. The maximum and minimum differences between the two wells were 52.1 m and 46.1 m, respectively. The mean and variance of well 1 were 0.99 and 0.15, and that of well 2 were 50.11 and 2.42, respectively. The mean and variance of the two wells were 49.12 and 2.27, respectively. The skewness and kurtosis of well 1 were 0.57 and 3.17, respectively. The skewness and kurtosis of well 2 were 0.03 and 2.54, respectively. The skewness and kurtosis of well 1 and well 2 are 0.54 and 0.63, respectively. The groundwater level time series diagram and histogram of the two wells are shown in Figure 2. Figure 2a shows daily time series of groundwater levels of two wells; it is evident that there is an increasing trend in groundwater level of well 2, and there is a little fluctuation of trend in groundwater level of well 1. Figure 2b,c shows that the groundwater level of the two wells both follow a normal distribution.

2.2. Partial Mutual Information (PMI)

Mutual information (MI) represents the amount of information contained in one random variable about another random variable [39]; in other words, $MI$ quantifies the interdependence between two random variables [40]. It is used to select the input of the model by calculating the mutual information value of input variable and output variable [21]. Assuming $X$ and $Y$ are two discrete random variables, the partial mutual information ($PMI$) is an improvement of the mutual information method [41]. Compared with $MI$, $PMI$ measures the additional correlation between the new alternative input variables and the forecast object in the case of eliminating the correlation of the selected input variables, and effectively avoids the inclusion of redundant variables [42]. PMI is defined as:

$$PMI=\iint f_{X^{\prime },Y^{\prime }}(x^{\prime },y^{\prime })\ln [\frac{f_{X^{\prime },Y^{\prime }}(x^{\prime },y^{\prime })}{f_{X^{\prime }}(x^{\prime })f_{Y^{\prime }}(y^{\prime })}]d{x}^{\prime }d{y}^{\prime }$$

(1)

$$x^{\prime }=x-E[x|z];y^{\prime }=y-E[y|z]$$

(2)

where E represents expected value; x represents the alternative input variable; $y$ represents the forecast object; $z$ represents the set of selected forecast variables; $y^{\prime }$ represents the residuals that exclude z from affecting $x$; $y^{\prime }$ represents the residuals that exclude z from affecting $y$. To estimate the marginal probability density function and joint probability density function of variables in partial mutual information, Gaussian function is adopted as the kernel function to estimate the sample probability density:

$${\widehat{f}}_X(x_i)=\frac{1}{N}{\displaystyle \sum _{j=1}^N\frac{1}{{(2\pi )}^{d/2}\lambda {(detS)}^{1/2}}}\times exp(\frac{-{(x_i-y_j)}^T{S}^{-1}(x_i-y_j)}{2{\lambda }^2})$$

(3)

where ${\widehat{f}}_x$ represents the estimate of the density function of $X$ at $x_i$; d is the dimension of $X$; $S$ is the covariance matrix of $X$; $detS$ is the determinant of $S$; $\lambda$ is the window width of kernel density estimation, the empirical formula is:

$$\lambda ={(\frac{4}{d+2})}^{1/(d+4)}{N}^{{}^{-1/(d+4)}}$$

(4)

2.3. Long Short-Term Memory (LSTM)

LSTM model is a recursive model used to solve the long-term dependence problem of a traditional recurrent neural network (RNN) [43,44]. The LSTM model is composed of input gate, memory cell, forget gate, and output gate, as shown in Figure 3. The forget gate specifies which information is deleted from the cell state, the input gate specifies which information is added to the cell state, and the output gate specifies which information in the cell state is used [45,46]. The gate structure can maintain or adjust the state of the storage unit, and the main features of the model are contained in the hidden layer composed of the storage unit. In each memory unit, there are three input information points at time t: the input $X_t$ at the current time, the memory unit state $C_{t-1}$ of the previous time t−1, and the output $h_{t-1}$ of the storage unit at the previous time t−1. There are two output information points: the output $h_t$ at the current time t and the unit state $C_t$ at the current time. The internal structure of LSTM memory unit at time t and its correlation with the state of adjacent time memory units: the specific calculation process of a single memory unit at time t is as follows. Determine the fate of the information before time t, and the unnecessary information will be deleted in this step through the forget gate. The forget gate adopts sigmoid activation function σ [47]. The forget gate f_t is calculated as shown in the formula:

$$f_t=\mathsf{\sigma }\left(W_f\cdot [X_t,h_{t-1}]+b_f\right)$$

(5)

where $W_f$ represents the weight matrix of forget gate; $b_f$ represents the bias of the forget gate.

Determining the fate of the new input information and updating the unit status consists of two parts. Firstly, the sigmoid activation function determines whether to update or ignore the new information. Secondly, the tanh function gives weight to the passed value to determine its importance level. The input gate is calculated as shown in the formula:

$$i_t=\mathsf{\sigma }\left(W_f\cdot [X_t,h_{t-1}]+b_i\right)$$

(6)

Next, $h_{t-1}$ and $X_t$ are used to calculate the cell state for the current input, as shown in the equation:

$$\widetilde{C_t}=tanh\left(W_c\cdot [X_t,h_{t-1}]+b_c\right)$$

(7)

Next, $C_{t-1}$ and $f_t$ are used to calculate the cell state, $C_t$ as shown in the formula:

$$C_t=f_t*C_{t-1}+i_t*{\tilde{C}}_t$$

(8)

where $W_i$ and $W_c$ represent the weight matrix of the input gate and the calculation unit, respectively; $b_c$ represents the offset term of the calculation unit.

Calculate the hidden layer state of the memory unit at time t. Firstly, $O_t$ in the forget gate determines which part of the cell state is output by the activation function sigmoid, as shown in the formula:

$$O_t=\mathsf{\sigma }\left(W_o\cdot [X_t,h_{t-1}]+b_o\right)$$

(9)

where $W_o$ represents the weight matrix of output gate; $b_o$ represents the offset term of the output gate. Next, the final output of LSTM is calculated by $tanh(C_t)$ and the output of $O_t$, as shown in the formula:

$$h_t=O_t*tanh(C_t)$$

(10)

2.4. Bootstrap

Bootstrap method is a statistical inference method of expanding samples [48]. Assuming that a given observation sample is a totality, the overall distribution is unknown. There are random samples that are put back in totality, and the resulting new samples are called self-help samples. Through repeated sampling of the totality, the estimated values of parameters such as mean and variance are calculated for each self-help sample [49]. According to the distribution histogram of corresponding estimated parameters, the empirical distribution of parameters estimated by the bootstrap method can be obtained, which is generally assumed to be normal distribution. Using this information to estimate the distribution parameters of the totality, the statistical parameters of the distribution function of the observation samples are obtained, and the corresponding estimation parameters such as mean and variance are calculated. The advantage of bootstrap is that resampling is only based on observation samples, and there is no need to assume the overall distribution. With the help of bootstrap sampling, the uncertainty of distribution parameter estimation may be quantitatively estimated [50,51] due to the limitation of data length.

2.5. Prediction and Uncertainty Evaluation Metrics

In this study, the coefficient of determination ($R^2$) and root mean square error ($RMSE$) are used to evaluate the accuracy of the model. The specific calculation formulae are as follows:

$$R^2=1-\frac{{\displaystyle \sum _{i=1}^n{({\widehat{y}}_i-y_i)}^2}}{{\displaystyle \sum _{i=1}^n{({\widehat{y}}_i-{\overline{y}}_i)}^2}}$$

(11)

$$RMSE=\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n{({\widehat{y}}_i-y_i)}^2}}$$

(12)

where $y_i$ is the observed value of daily GWL, m; ${\widehat{y}}_i$ is the predicted value of GWL, m; ${\overline{y}}_i$ is the mean value of GWL, m. The range of $R^2$ is between 0 and 1. The closer $R^2$ is to 1 or $RMSE$ is to 0, the better the accuracy of the model is.

Three uncertainty evaluation metrics are used to evaluate the uncertainty, including coverage rate ($CR$), relative width ($RB$), and relative offset degree ($RD$). The formula is as follows:

$$CR=\frac{n}{N}$$

(13)

$$RB=\frac{1}{N}\cdot {\displaystyle \sum _{i=1}^N\frac{(l_i^u-l_i^l)}{L{}_{sim}{}^i}}$$

(14)

$$RD=\frac{1}{N}{\displaystyle \sum _{i=1}^N(\left|\frac{1}{2}(l_i^u+l_i^l)-L{}_{obs}{}^i\right|/L{}_{sim}{}^i)}$$

(15)

where $L{}_{obs}{}^i$ and $L{}_{sim}{}^i$ are the observed and predicted GWLs at the moment $i$, respectively; $l_i^u$ and $l_i^l$ are the upper and lower limits of the corresponding uncertainty interval at the moment $i$, respectively; $n$ is the number of observed values within the uncertainty interval; and $N$ is the total number of observed values.

3. Results and Discussion

3.1. Predictors Selection

There are ten teleconnection patterns information, namely Antarctic Oscillation (AAO), Arctic Oscillation (AO), Southern Oscillation Index (SOI), Pacific North American Index (PNA), North Atlantic Oscillation (NAO), sunspot (SN), East Central Tropical Pacific SST (Niño 3.4), Extreme Eastern Tropical Pacific SST (Niño 1 + 2), Central Tropical Pacific SST (Niño 4), Eastern Tropical Pacific SST (Niño 3), and two variables including antecedent precipitation (P) and antecedent groundwater level (AGWL). Each variable with six-lagged days was considered, and there were 72 candidate predictors in fact. Partial mutual information (PMI) can be effective for predictor selection without redundant variables.

For well 1, the order of determined predictors was AGWL (t-1) (GWL with 1-day lagged), AGWL (t-2), AGWL (t-3), AGWL (t-4), AGWL (t-5), AO (t-3), Niño 3.4 (t), P (t), Niño 3.4 (t-1), Niño 1 + 2 (t-2), Niño 1 + 2 (t-1), Niño 3.4 (t-2), Niño 1 + 2 (t), Niño 3.4 (t-3). For well 2, the order of determined predictors was AGWL (t-1), AGWL (t-5), AGWL (t-2), AGWL (t-4), AGWL (t-3), Niño 3.4 (t-5), Niño 3 (t-5), AO (t), Niño 3 (t), Niño 1 + 2 (t-1), PNA (t), SN (t-5). According to PMI results, groundwater level in well 1 showed a significant correlation with AGWL, AO, Niño 3.4, Niño 1 + 2, and antecedent precipitation, but groundwater level in well 2 showed a significant correlation with AGWL, AO, Niño 3.4, Niño 3, Niño 1 + 2, and PNA. Antecedent GWL plays the most significant roles in GWL prediction, and AO as teleconnection patterns has a great impact on the GWL of the two wells. Precipitation is a significant predictor in the humid areas, no precipitation is a significant predictor in the arid areas.

3.2. Groundwater Level Prediction

3.2.1. LSTM Development

All the parameters of LSTM models including the number of hidden units, learn rate, batch size, and dropout were set though a trial-and-error procedure. The learning rates and the number of hidden units plays a more important role in the performance of LSTM models. The small learning rate means LSTM model takes a long time to convergence state, and a large learning rate may result in a large fluctuation of the loss and difficult to achieve converge.

The learning rates and the number of hidden units should be determined according to the corresponding R² values. The learning rates were set from 0.1 to 1 and the number of hidden units was set from 10 to 100. Results of different values of learning rates and the number of hidden units on the accuracy of the LSTM models and the optimal parameters of the LSTM models are shown in Figure 4. From Figure 4, for well 1, the highest R² can reach to 0.97 when the learning rates and the number of hidden units are 0.9 and 70, respectively. For well 2, the highest R² can reach 0.72 when the learning rates and the number of hidden units are 0.8 and 30, respectively.

3.2.2. Comparison of LSTM and GRNN

Generalized regression neural network (GRNN) is a kind of radial basis neural network, which has strong nonlinear mapping ability and learning speed, and has stronger advantages than radial basis function neural network (RBFNN). Finally, the network generally converges to the optimized regression with large sample size. When the sample data are small, the prediction effect is good, and the unstable data can also be processed. Although GRNN seems to be less accurate than radial basis function, it actually has great advantages in classification and fitting. More detailed information can be found in some studies [52,53].

In this study, GRNN and LSTM models were used to compare the model performance in terms of RMSE and R².

Table 2. The performance of MLR, GRNN, SVR, and LSTM for high flow during the calibration and validation period.

Model	Well 1				Well 2
	Calibration		Validation		Calibration		Validation
	RMSE	R2	RMSE	R2	RMSE	R2	RMSE	R2
GRNN	0.17	0.92	0.16	0.87	0.64	0.90	1.43	0.54
LSTM	0.15	0.98	0.12	0.97	0.45	0.95	1.14	0.72

shows the performance of GRNN and LSTM for two wells during calibration and validation period. As shown in Table 2, for the calibration stage of well 1, the RMSE and R² of LSTM are the minimum and maximum values of two models, with a value of 0.15 and 0.98, respectively. Compared with GRNN, the RMSE of LSTM decreased by 11.76%, the R² of LSTM increased by 6.52%, respectively. For the validation period, RMSE and R² of LSTM were also 0.12 and 0.97, respectively. Compared with GRNN, RMSE decreased by 0.04, and R² increased by 0.1. For the calibration period of well 2, the RMSE and R² of LSTM were also the minimum and maximum values of two models, 0.45 and 0.95, respectively. For the validation period, the RMSE and R² of LSTM were 1.14 and 0.72, respectively. Compared with GRNN, RMSE decreased by 20.28%, the R² increased by 33.33%.

Figure 5 and Figure 6 show the scatter plot of observed vs. predicted groundwater level for two models during the calibration and validation period. As shown in Figure 5, in the calibration period, the fitting performance of predicted value and observed value of LSTM in well 1 is better than GRNN. The fitting performance of GRNN in well 2 is similar to LSTM, but the performance of LSTM is obviously better than GRNN at low or high groundwater levels. The similarity can be found in Figure 6. In general, LSTM has better performance than GRNN.

3.2.3. Comparison of Different Predictor Scenarios

Four predictor scenarios were used to analyze the influence of the predictors on the accuracy of LSTM models for groundwater level prediction, and they include: (1) antecedent precipitation (P), (2) P and antecedent groundwater level (AGWL), (3) P, AGWL, and teleconnection (T), and (4) predictors selected by PMI (PBP).

As shown in

Table 3. The accuracy of LSTM with different predictors scenarios for two wells during calibration and validation period.

Scenarios	Well 1				Well 2
	Calibration		Validation		Calibration		Validation
	RMSE	R2	RMSE	R2	RMSE	R2	RMSE	R2
Scenarios 1	0.26	0.71	0.23	0.70	0.73	0.87	1.47	0.50
Scenarios 2	0.20	0.86	0.19	0.78	0.45	0.95	1.31	0.60
Scenarios 3	0.18	0.90	0.18	0.82	0.41	0.96	1.24	0.64
Scenarios 4	0.15	0.98	0.12	0.97	0.40	0.96	1.14	0.72

, during the calibration and validation period, for well 1, the RMSE and R² of the model with the predictors selected by PMI (PBP) were the minimum and maximum values among four scenarios, which were 0.15 and 0.98, and 0.12 and 0.97, respectively. The prediction effect of well 2 under scenario 4 is the same as that of well 1, which is the best among the four scenarios. RMSE and R² are 0.40 and 0.96, and 1.14 and 0.72, respectively.

Figure 7 corresponds to Table 3 and shows the observed and predicted groundwater level under different scenarios. Compared with the other three scenarios, the fitting performance of the prediction factors of well 1 and 2 selected by PMI (PBP) is significantly closer to the observed values. Therefore, the predictors scenarios selected by PMI (PBP) had the best performance. The selection process is not only conducive to avoiding the selection of redundant variables, but also has a strong screening ability for the factors of nonlinear correlation.

3.3. Uncertainty Evaluation

As shown in

Table 4. Uncertainty evaluation of groundwater level prediction by LSTM and GRNN model.

	Calibration			Validation
	CR	RB	RD	CR	RB	RD
Well 1-GRNN	0.92	0.35	0.15	0.90	0.11	0.10
Well 1-LSTM	0.99	0.32	0.07	0.97	0.04	0.05
Well 2-GRNN	0.92	3.57	0.33	0.64	5.48	0.60
Well 2-LSTM	0.96	3.52	0.09	0.81	3.69	0.09

, for well 1, the CR values of LSTM in calibration period and verification period were 0.99 and 0.97 and closer to 1. Compared with GRNN, both CR values increased by 0.07. This suggests that the LSTM model’s predictions are even better. The CR values of well 2 in the calibration and verification period were 0.96 and 0.81, respectively. Compared with GRNN, CR increased by 0.04 and 0.17. This shows that compared with GRNN, more prediction results using LSTM model were within the confidence interval. For RB and RD, the LSTM value of well 1 were smaller than that of GRNN, which were 0.32 and 0.07, and 0.04 and 0.05, respectively. The above shows that the LSTM model is reliable in predicting the groundwater levels of the two wells.

Figure 8 shows the confidence interval compared to observations for two wells. In order to see the results of the confidence interval more clearly, this study chooses to focus on the results from 2022/1/11 to 2022/4/11. As can be seen from the figure, most of the observed values of well 1 were distributed between confidence intervals, and only a few were lower than lower quartile but not lower than the Min. All the observations of well 2 were basically within the confidence interval. The results show that the LSTM model can reliably predict the groundwater level of the two wells.

4. Conclusions

In this paper, LSTM coupled with PMI and bootstrap is introduced for groundwater level prediction and uncertainty for two wells in Iowa and Arizona, USA. PMI was used for identifying significant predictors for GWLs prediction models of the two wells. According to PMI results, groundwater level in well 1 showed a significant correlation with AGWL, AO, Niño 3.4, Niño 1 + 2, and antecedent precipitation, but the groundwater level in well 2 showed a significant correlation with AGWL, AO, Niño 3.4, Niño 3, Niño 1 + 2, and PNA. The significant predictors were employed for the inputs of LSTM models for GWL prediction, antecedent GWL played the most significant role in GWL prediction, and AO as teleconnection patterns had a great impact on the GWL of two wells. There was precipitation as a significant predictor in the humid areas and no precipitation as a significant predictor in the arid areas. The optimal LSTM model structure and parameter values were determined according to the accuracy of LSTM models for two wells. The prediction results were compared with GRNN models. It was concluded that LSTM with R² 0.97 and 0.72 performed superiorly to GRNN model with R² 0.87 and 0.54 for two wells. The performance of LSTM was improved by increasing correlation coefficient (R²) values by 10% and 25% for two wells compared to GRNN models. The LSTM model achieved higher accuracy in humid areas, while the performance in arid areas was poorer probably due to limited precipitation information. The influence of the predictors on the accuracy of LSTM models for two wells was analyzed, and the accuracy of LSTM with the predictors selected by PMI was found to be superior to that with all variables. The predictor selection can assist in improving the prediction performance of LSTM model. Finally, the prediction uncertainty for two wells was evaluated using LSTM integrated with bootstrap. The results indicate that LSTM reduces the uncertainty compared to GRNN model by providing a higher value of coverage rate (CR) and a lower value of relative width (RB) and relative offset degree (RD). LSTM coupled with PMI and bootstrap not only gave an accurate groundwater level prediction, but also evaluated the uncertainty of groundwater level prediction. However, the performance of groundwater level prediction in arid areas is not good enough, more hybrid models integrating wavelet transform, Box–Cox transformation, and LSTM models should be further studied to improve the performance of groundwater level prediction in different areas with more complex climate characteristics, such as coastal areas and underground reservoir management areas.