A new groundwater depth prediction model based on EMD-LSTM Open Access

Groundwater depth is a complex, random and fuzzy hydrological change. Especially in the case of unconfined aquifers, it is not only affected by various factors such as rainfall, evaporation and mining volume, but also affected by human activities. It is a typical weakly dependent and complex system (Wang et al. 2005; Chai et al. 2006; Jeong & Kim 2009; Chen et al. 2018). The accuracy of the prediction model of groundwater depth is of great significance for the rational utilization of water resources, the sustainable development of the regional social economy, and the evaluation of the ecological environment (Zhang et al. 2019). The study of the groundwater depth prediction model has always been one of the hot issues in the field of water conservancy at national and international level. In addition, domestic and foreign scholars have carried out a lot of research on the prediction of groundwater depth and achieved fruitful results. Pratim & Dash (2014) used IDW (inverse distance weighting), RBF (radial basis function), OK (ordinary kriging) and UK (universal kriging) interpolation methods to predict the temporal and spatial variation of groundwater depth. Pradhan et al. (2018) studied the application effect of soft computing technologies such as CANDIS, fuzzy logic and RBFN (radial basis function network) in the prediction of groundwater levels in the study area. Ghose et al. (2010) used two neural network models to analyze and compare the depth of groundwater. Zhao et al. (2016) constructed a classification regression tree model of groundwater level and applied it to groundwater level prediction. Yang & Zhang (2020) proposed the modeling steps and solution ideas of the improved non-parametric time series model in the dynamic prediction of groundwater levels. The Sun et al. (2021) prediction of groundwater depth in Zhongmu county was based on a multivariate time series model. Shen et al. (2006) used the grey memory model to predict the depth of groundwater in Hotan, Xinjiang. Zhang et al. (2022) used five prediction models of multiple linear regression, grey GM (1,1) (grey models), grey GM (1,1) based on Markov chain optimization, BP neural network, and BP neural network based on genetic algorithm optimization to predict groundwater depth. From the above, it is known that there are many methods for domestic and foreign researchers to predict groundwater depth, mainly focusing on regression analysis of groundwater depth time series and the use of fuzzy logic, interpolation methods and neural networks. Common algorithms such as RNN (recurrent neural network) have the problem of gradient disappearance or gradient explosion. LSTM have been used to solve the long-term dependency problem that is common in general recurrent neural networks since the beginning of their design. Using LSTM can effectively transfer and express long-term sequences information and does not cause useful information from a long time ago to be ignored. At the same time, LSTM can also solve the gradient vanishing/exploding problem in RNN. EMD decomposes the time series empirical mode into a set of IMF (intrinsic mode functions), which were combined into a time–frequency amplitude spectrum after the Hilbert transform, which is used to represent the characteristics of the signal. This method can analyze both linear steady-state signals and nonlinear non-steady-state signals. Although both EMD decomposition and neural network are applied in groundwater depth, it is rare to use the combination of EMD and LSTM network to construct a coupled model for groundwater depth prediction. Combining the advantages of EMD and LSTM neural networks, the paper establishes a coupled model of groundwater depth prediction based on EMD and LSTM networks.

EMD (Huang et al. 1998) is used to decompose the signal according to the time scale characteristics of the data itself and no basis function needs to be set in advance. This is because of the characteristic that the EMD method can theoretically be applied to the decomposition of any type of signal, so it has obvious advantages in processing non-stationary and nonlinear data and is suitable for analyzing nonlinear and non-stationary signal sequences with a high signal-to-noise ratio.

Groundwater depth is deeply affected by a variety of influencing factors and its changes are random, uncertain, and fluctuating. The EMD method is selected in the research because it can decompose the complex signal into a finite number of IMF and the decomposed IMF components contain local characteristic signals of different time scales of the original signal. It can smooth the non-stationary data, and then perform the Hilbert transform to obtain the time spectrum and obtain the frequency with physical meaning, thereby reducing the prediction error. The IMF must meet the following two assumptions: (1) the total number of extreme points and zero-crossing points in the IMF must be equal or have a maximum difference of 1 (Cai et al. 2010); (2) at any time, the extreme value of the IMF must be the upper envelope formed by large values and the lower envelope formed by minimal values must have an average value of zero.

The implementation steps of EMD decomposition are as follows (Xu et al. 2016):

Step 1: Find the maximum and minimum values of the time series {} and fit all the extreme points by the cubic spline interpolation function to obtain the upper envelope {} and the lower envelope {}.

In the formula: is upper envelope and is lower envelope.

Step 4: Check whether is an IMF, that is, whether the number of extreme points is equal to the number of zero points or the difference between the numbers cannot exceed one at most and the average value of the upper and lower envelopes at any time is zero. If is taken as the first IMF is recorded as ⁠; otherwise, is used as a new input sequence and steps 1–4 are repeated until the IMF is obtained.

LSTM is a special form of recurrent neural network that introduces three gate units, forgetting gate, input gate, output gate and memory cells (Zheng et al. 2017). This well designed structure alleviates the vanishing gradient problem of RNN (Hochreiter & Schmidhuber 1997). The structure of the LSTM neuron is shown in Figure 1, is the current time input, and are the hidden states of the previous and current moments, respectively; ⁠, ⁠, and are the input gate, forgetting gate, and output gate at the current moment, respectively; is the candidate memory cell at the current moment; ⁠, are the memory cells at the previous and current moment, respectively, and is the output.

From the perspective of EMD decomposition, the contribution rates of the IMF components and residuals to the groundwater depth sequence are not the same, and the IMF components and residuals can be approximately regarded as the driving factors of groundwater depth. Then the prediction of groundwater depth is equivalent to the prediction of IMF components and residuals. The specific steps of the coupling model of groundwater depth prediction model based on EMD and LSTM network are as follows:

• (1)

  The EMD decomposition of the groundwater depth series from 2005 to 2021 is performed using MATLAB to obtain the IMF components and residuals of the groundwater depth series.

• (2)

  Standardize the IMF components and residuals of the groundwater depth series.

If the range of the input or output data of the network is quite different, the prediction model of the network will have a large error, so we must standardize the data so that the data range is within [0, 1].

• (3)

  The IMF components and residuals of the groundwater depth from 2005 to 2019 are used as the training data of the LSTM network, and the IMF components and residuals of the 2020–2021 years are used as the prediction data of the LSTM network.

• (4)

  Using the LSTM network to predict the IMF components and residuals of groundwater depth from 2020 to 2021.

• (5)

  Finally, the predicted groundwater depth IMF components and residuals are accumulated and restored according to the formula.

Root mean square deviation: This is the square root of the squared deviation of the observed value from the true value and the ratio of the number of observations n. In actual measurement, the number of observations n is always limited, and the true value can only be replaced by the most reliable (best) value. Root error is very sensitive to very large or very small errors in a group of measurements, so the root mean square error can well reflect the precision of the measurement.

Relative Error: refers to the value obtained by multiplying the ratio of the absolute error caused by the measurement to the measured (conventional) true value by 100%, expressed as a percentage. Generally speaking, the relative error is a better indicator of the confidence level of the measurement.

– actual relative error, usually given as a percentage

– absolute error

– truth value.

Zhaogang town is located in the northeast of Fengqiu county, Xinxiang city. The town has a total area of 79.15 square kilometers. The region has a warm temperate continental monsoon climate, with an annual average temperature of 13.5 °C–14.5 °C, annual precipitation of 615.1 mm, and the frost-free period of 214 days. Chengguan town is located in the North Henan Plain, facing the Yellow River to the south, with a total area of 58 square kilometers. The region has a warm temperate continental monsoon climate, the annual average temperature is between 14.6 °C and 15.3 °C, the annual precipitation is 603.5 mm, and the frost-free period is 226 days. Zhaogu town is located in the southwest of Zhaogu town, Weihui county, Xinxiang city with a total area of 60.38 square kilometers, the annual precipitation is 598 mm, and the frost-free period is 203 days. Through comprehensive and scientific monitoring of groundwater in three areas, the dynamic characteristics of groundwater are studied, the temporal and spatial evolution of groundwater is understood and its dynamic change characteristics are mastered. On this basis, groundwater depth is predicted and groundwater level changes are analyzed. The trend can provide a theoretical basis for the sustainable utilization of groundwater resources, the safety of the ecological environment, and the sustainable and healthy development of society and economy, and provide decision-making for administrative departments at all levels to formulate groundwater resource planning, agricultural development planning, ecological environmental protection and governance, and social and economic development planning. The data in this paper are obtained from the monitoring data of No. 5 well in the northwest of Sun village, Zhao Gang town, Fengqiu county, Xinxiang city, No. 19 well in the west of Bali village, Chengguan town, Yuanyang county, Xinxiang city, and No. 19 well in the east of Zhaogu town, Weihui county, Xinxiang city, from 2005 to 2021. These three wells are professional observation wells, which can fully reflect the real dynamic changes of groundwater, with good representation of monitoring data and meet the technical requirements of groundwater monitoring specifications.

It can be seen from Figure 2 that from 2005 to 2021, the groundwater depth of each professional observation well generally showed an upward trend. The amplitudes are inconsistent, which also verifies that the groundwater depth is uncertain and non-stationary. At the same time, it also reflects that the EMD method is reasonable from the side.

Following the steps of the previous EMD decomposition, the EMD decomposition is performed for the groundwater depth data of well 5 in Zhaogang town, well 19 in Chengguan town, and well 19 in Zhaogu town from 2005 to 2021.

As can be seen from Figures 3–5, the groundwater depth sequence is decomposed into five IMF components and a corresponding residual. Among them, the first IMF component has the largest volatility, high frequency, and shortest wavelength; other IMF components gradually decrease in amplitude, decrease in frequency, and increase in wavelength. The volatility and non-smoothness of the groundwater depth series of three professional observation wells are greatly reduced after EMD processing.

When using the LSTM network to predict the groundwater depth, it is necessary to divide the training and test samples. The IMF and residual data from 2005 to 2019 are used as training samples, and the IMF and residual data from 2020 to 2021 are used as test samples.

Through a large number of experiments, it was found that LSTM has 200 hidden units to be optimal. To prevent the gradient from exploding, we set the gradient threshold to 1, specified the initial learning rate of 0.005, and reduced the learning rate by a multiplication factor of 0.2 after 125 rounds of training.

According to the previous steps, the LSTM network was used to predict the five IMF components and one residual of the groundwater buried depth of Well No. 5 in the northwest of Sun village, Zhaogang town, Fengqiu county, Xinxiang city from 2020 to 2021. The prediction results are shown in Figure 6.

It can be seen from Figure 6 that the prediction effect of the IMF1 component is slightly worse, which indicates that the non-stationarity of the IMF1 component is higher; the prediction effect of IMF2 to IMF5 is better. This shows that the non-stationarity of IMF2 to IMF5 is lower. After the groundwater depth sequence is decomposed by LSTM, the volatility and non-stationarity of the sequence are greatly reduced.

It can be seen from Table 1 that the RMSE value of IMF1 is large, which is 0.11512, and the RMSE values of IMF2 to IMF5 and residuals are all small, which are 0.01208, 0.03655, 0.01185, 0.00898, and 0.0004, respectively. And it can be seen that the maximum relative errors of IMF1 to IMF5 are relatively large, which are 506.78%, 393.41%, 150.38%, 189.23%, and 96.05%, respectively. The minimum relative error and average relative error of IMF1 are also large, which are 21.39% respectively, 311.46%. This shows that the non-stationarity of the IMF1 component is higher, which has a greater impact on the prediction error; the minimum relative errors of IMF2 to IMF5 and residuals are all small, 0.2%, 0.32%, 0.26%, 0.05%, and 0.0002%, respectively. The average relative error of IMF2 is 41.88% larger. The prediction effect of residuals is the best, with the maximum relative error, minimum relative error, and average relative error being 0.01%, 0.0002%, and 0.00004%, respectively. It can be seen from Table 1 that after the EMD decomposition of the groundwater depth sequence, the IMF component becomes more stable, and the IMF1 to IMF5 and the RMSE indicators of the residual show a decreasing trend as a whole. Although some points in IMF1 to IMF5 have large errors, the proportion of individual points in the sequence of groundwater depth is very small, so it will not affect the overall error of groundwater depth.

It can be seen from Table 2 that the maximum, minimum and average relative errors of the EMD-LSTM coupled prediction model are 2.00%, 0.07%, and 1.18%, respectively. The relative error of model prediction is small, and the pass rate is high.

Figure 7 is the prediction curve of groundwater depth from 2020 to 2021 in Well No. 5 in the northwest of Sun village, Zhaogang town, Fengqiu county, Xinxiang city. The predicted value of groundwater depth in 2021 is consistent with the actual value.

Using this method as an example, predictions were made for Well No. 19 west of Bali village, Chengguan town, Yuanyang county, Xinxiang city, and Well No. 19 east of Zhaogu town, Weihui county, Xinxiang city. The prediction results are shown in Figures 8 and 9.

It can be seen that the predicted value is basically consistent with the actual value. The fitting degree of the EMD-LSTM coupled model is high.

Tables 3–5 show the comparison results of the prediction errors between the EMD-LSTM model and other models.

It can be seen from Tables 3–5 that the qualified rate of the EMD-LSTM coupling model for the prediction of groundwater depth is 100%, and the relative error is low. The model is obviously better than the single LSTM network model and the BP model. The LSTM network has a large period, and the network is very deep, the calculation amount will be large, it is time-consuming, and the BP network cannot learn well for some high-frequency data so that the prediction accuracy is improved.

In this paper, the EMD decomposition and the LSTM neural network model are combined to establish the EMD-LSTM neural network model and apply it in three professional observation wells. The reliability of the results was compared with the results of the commonly used BP neural network model and a single LSTM model, and the following conclusions were drawn:

• (1)

  The randomness, uncertainty, and volatility of groundwater depth series are greatly reduced after EMD decomposition, and the interference and non-stationarity of characteristic information at different scales are reduced, which effectively improves the prediction accuracy of the model. The EMD-LSTM coupling model predicts that the overall prediction effect is good, and the relative error and absolute error are small. The model pass rate is 100%, which is better than a single LSTM, BP neural network. This shows that the EMD-LSTM coupled model is feasible for groundwater depth prediction.

• (2)

  The time series of groundwater depth is decomposed by EMD, and the signal is decomposed into several IMF components and residuals, and the predicted value is equal to the sum of the predicted values of several IMF components and residuals. Although some IMF components have relatively large prediction errors, these IMF components account for less in the entire signal. When the predicted values of the IMF components and residuals are converted into the overall predicted values, the overall error will be weakened.

• (3)

  Although the EMD-LSTM coupling model has high accuracy in the prediction of groundwater depth, the prediction accuracy of high-frequency components is relatively poor. In addition, the study only makes short-term predictions of groundwater depth but does not make long-term predictions, and the model did not involve the physical mechanism of groundwater depth evolution. However, by decomposing the original sequence into sub-components in each frequency domain, the predicted value of the original sequence is equal to the sum of the predicted values of the sub-components in the frequency domain. This decomposition–reconstruction prediction model can also be used for reference to other aspects of time series data prediction.

This work was supported by the Key Scientific Research Project of Colleges and Universities in Henan Province (CN) [grant numbers 17A570004].

All authors contributed to the study conception and design. writing and editing: Xianqi Zhang and Haiyang Chen; chart editing: Dong Zhao; preliminary data collection: Guoyu Zhu, Bingsen Duan. All authors read and approved the final manuscript.

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All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.