Wavelet analysis–artificial neural network conjunction models for multi-scale monthly groundwater level predicting in an arid inland river basin, northwestern China

Groundwater is a valuable water resource for public water supplies, agriculture, industries, communities, and the well-being of natural ecosystems in arid regions (Zhao et al. 2011; Wei et al. 2014; Wu et al. 2014). Arid regions face major challenges in the sustainable management of scarce groundwater resources, particularly due to external pressure of population increase, economic developments, climate change, pollution, and over-abstraction. Therefore, accurate and reliable groundwater level forecasts are particularly important for sustainable management of groundwater to meet human developments, irrigation scheduling, environmental planning, and ecosystem needs (Coulibaly et al. 2001; Brunner et al. 2009; Wu et al. 2010; Moosavi et al. 2013; Zhu et al. 2013; Gorgij et al. 2016).

Groundwater models are widely used for understanding dynamical changes in groundwater systems and their availability for proper decision-making in terms of water resource management (Banks et al. 2011; Bourgault et al. 2014; Li et al. 2015, 2016; Vainu et al. 2015). Generally, physical models are used for characterizing a groundwater flow system and predicting the groundwater level fluctuations. These models establish a governing equation to simplify the physics of the flow in the subsurface and solve it with proper initial and boundary conditions using numerical methods (Mohanty et al. 2015). Spatial and temporal groundwater level distributions can be simulated within a given domain. Although physical models are tools for depicting hydrological variables and understanding the hydrological processes that occur in a system, they do have practical limitations. Most models are complex because they need several different inputs that are not easily accessible. Moreover, the relationships between groundwater level fluctuations and hydrological variables such as precipitation, temperature, evaporation, river stage, and pumping are likely to be non-linear. Therefore, linearly based physical models often fail to effectively represent subsurface processes so as to make modeling a very expensive and time-consuming process (Jha & Sahoo 2015). Under data-scarce conditions that are common issues in most developing areas, physical models are highly restricted in their practical usage (Mohanty et al. 2015). A viable alternative, artificial neural network (ANN) model mimics the biological structure of the brain and possesses good flexibility in simulating non-linear processes in groundwater modeling (Coulibaly et al. 2001; Chang et al. 2015; Jha & Sahoo 2015; Mohanty et al. 2015).

In spite of the flexibility of a regular ANN model for its application in groundwater modeling, a significant challenge is its limitations with non-stationary data. Consequently, an ANN model is unable to handle non-stationary as well as seasonality in a hydrologic dataset if an appropriate pre-processing of the inputs is not performed (Nourani et al. 2014). Furthermore, if the input parameters possess trends or periodicities, such patterns are not utilized satisfactorily when simulating the groundwater time series. A common tool for pre-processing non-stationary data in hydrology is wavelet analysis (WA), which provides information about the frequency components of the input signal. As wavelet transformation decomposes input data into space–frequency components, it reveals a considerable amount of concealed data patterns, and therefore, provides insight into the physical aspects of input data (Dabuechies 1990). The information captured at various resolution levels of the inputs can, therefore, assist in utilizing this information for more accurate predictive modeling of the phenomenon in question (Seo et al. 2015). Therefore, wavelet transformation has been used as an effective mining tool in analyzing non-stationary data used for predictive modeling in water resources (Nourani et al. 2014).

Scientific research into the application of wavelet coupled models has grown in recent years with several applications of WA-ANN, wavelet–adaptive neuro-fuzzy inference system (WA-ANFIS), and wavelet–support vector machine (WA-SVM) for groundwater forecasting. Adamowski & Chan (2011) proposed a new method coupling the wavelet and ANN technique (WA-ANN) for 1-monthly ahead groundwater level forecasting, and found that the WA-ANN model provided more accurate groundwater level forecasts than ANN and ARIMA models. Kisi & Shiri (2012) investigated the ability of the WA-ANFIS for 1-, 2-, and 3-day ahead groundwater depth forecasting. The results revealed that the WA-ANFIS models performed better than the ANFIS models. Moosavi et al. (2013) compared the WA-ANN, WA-ANFIS, ANN, and ANFIS models for 1-, 2-, 3-, and 4-month ahead groundwater level forecasting and the results demonstrated that the WA-ANFIS models were more accurate than WA-ANN, ANN, and ANFIS models. Suryanarayana et al. (2014) applied the WA-SVM model to predict 1-month ahead groundwater level for three observation wells in the city of Visakhapatnam, India. The study found the WA-SVM model yielded better accuracy compared with SVM, ANN, and ARIMA models. In a review on applications of hybrid wavelet and Artificial Intelligence (wavelet-AI) based models in hydro-climatology, Nourani et al. (2014) stated that application of wavelet-AI in groundwater forecasting has remained rare, despite its application in several other sub-areas in hydrology. Based on its success elsewhere, it is advocated that the potential of wavelet-AI methods for groundwater modeling should be explored.

In the arid northwest region of China, water resources play a dominant role in the sustainable development of the regional economy (Wu et al. 2010; Zhao et al. 2011). In such environments, surface water resources are generally scarce and highly unreliable in terms of sustained availability, so that groundwater is the main water resource system. The Zhangye basin (Figure 1) is an important center for development of agriculture and industry. In the last few decades, with significant progression of economic developments, water demands and its sustainability have become much greater than ever, especially for agricultural purposes, so both surface water and groundwater have been over-exploited (Wei et al. 2014). In the 1970s, the annual exploitation of groundwater was approximately 0.39 × 10⁸ m³ year⁻¹, but by 1999 this had risen to 2.17 × 10⁸ m³ year⁻¹ (Wen et al. 2007). With increasing intensity of human activities, such as pumping of groundwater, irrigating agricultural land, and construction of dams and water conveyance systems, the groundwater dynamical processes have become complex, and groundwater cycle processes have undergone significant aberrations (Wu et al. 2010; Zhao et al. 2011; Moiwo & Tao 2014; Cai et al. 2016). Therefore, in order to better understand the complex behavioral change in the groundwater environment and utilize groundwater resources effectively in this complex aquifer, it is necessary to develop accurate and reliable predictive models to forecast groundwater level fluctuations. However, for the catchment where future information of the aquifer exhibits complex behaviors in terms of groundwater dynamical processes, the application of WA-ANN models, whose performance has been shown to be superior to classical models elsewhere, is an interesting research endeavor. Also, importantly, the published literature shows that very few studies, if any, have explored the performance of WA-AI models for groundwater level forecasting, and in particular, that no research has explored these models for groundwater level forecasting with lead times greater than 1 month in the aquifer of Zhangye basin, China.

The novelties of this study are two-fold. (1) To develop a WA-ANN model for simulation of 1-, 2-, and 3-month ahead groundwater levels in the complex aquifer of Zhangye basin, northwestern China. Therefore, a multiscale predictive model is developed for short- (1 month) and medium-term (2–3 months) forecasts to facilitate an understanding of regional water resource systems. (2) To analyze the comparative performance of ANN and WA-ANN models in terms of their effectiveness in simulating groundwater levels.

ANN is a large parallel distributed information processing system that has certain structural characteristics resembling the biological neural networks of the human brain (Haykin 1999). Conceptually, a neural network is characterized by its inter-linked neurons and architecture that represent the pattern of connection between nodes, the connection weights and the activation function. The most commonly used neural network structure is the feed-forward hierarchical architecture. A typical three-layered feed-forward neural network comprises multiple elements also called nodes, and connection pathways that link them (Haykin 1999). The nodes are processing elements of the network and are normally the neurons, reflecting the fact that a neural network model is based on the biological neural network of the human brain. A neuron receives an input signal, processes it, and transmits an output signal to other interconnected neurons.

where J is the Jacobian matrix calculated using standard backpropagation techniques. The J contains first derivatives of network errors with respect to the weights and biases and e is a vector of errors. Notably, the Levenberg–Marquardt algorithm has been found to be significantly faster and efficient (Tiwari & Adamowski 2013; Deo & Şahin 2015). The sigmoid and linear activation functions were used for the hidden and output node(s), respectively.

where * indicates the conjugate complex function. By smoothly varying both s and τ, one can construct a two-dimensional picture of wavelet power, W(τ, s), showing the frequency (or scale) of peaks in the spectrum of x(t), and how these peaks change with time (Drago & Boxall 2002).

where W_m,n is the wavelet coefficient for the discrete wavelet of scale s = 2 m and location τ = 2 mn; N is an integer power of 2, N = 2 m.

In the DWT process, the original inputted time series is passed through the high-pass and low-pass filters in order to extract the high and low frequency components of the feature datasets. Consequently, the detailed wavelet coefficients and approximation series are obtained with the wavelet algorithm. The filtering step is repeated each time. A portion of the signal corresponding to certain frequencies is eliminated, obtaining the approximation and retaining details to be used as input for the wavelet-based model.

The study region, Zhangye basin, located in the middle reach of the Heihe River, covers an area of 1.08 × 10⁴ km² (Figure 1). This area has an arid form of the continental climate with a mean annual temperature between 3 and 7 °C. The average annual precipitation ranges from about 50 to 150 mm year⁻¹. The average potential evaporation rate is about 2,000–2,200 mm year⁻¹ (Wu et al. 2010). Zhangye basin is also an important center for the development of agriculture and industry (Wu et al. 2010), and therefore, is a useful candidate region for development of predictive models for groundwater forecasting. For the last few decades, water demand in this area has increased and groundwater has been over-abstracted due to the rapid population increase and remarkable economic developments. Due to extensive use of surface water and over-abstraction of groundwater, groundwater levels have lowered and the quality of groundwater also impacted (Zhao et al. 2011).

Zhangye basin is underlain by Tertiary red molasses, and filled with large volumes of unconsolidated Quaternary sediments, to the depths of 300–500 m. The Quaternary basins can be divided into discrete geomorphologic units, including piedmont alluvial plain, alluvial plain, and desert. The sediments in the basins from south to north gradually change from coarse-grained gravel to medium- and fine-grained sand and silt. These sediments along with aeolian and lacustrine deposits form the main aquifers. In the southern part of Zhangye basin, the aquifer is formed from highly permeable cobble and gravel deposits with a thickness of 300–500 m. From the northern edge of this diluvial fan, the aquifer becomes confined or semi-confined, with a thickness of 100–200 m, comprising interbedded cobble, gravel, fine sand, and clay. Further north, the groundwater table becomes shallow. The rivers originating from the Qilian Mountains are the main source of recharge for the local aquifers. Groundwater in the basin flows generally from the piedmont area towards the center of the basin (Wen et al. 2007; Wu et al. 2010).

In this research, the time series of the monthly groundwater level records obtained from two observation wells in the Zhangye Basin, namely Well I and Well II, were selected (Figure 1). Accordingly, the climatic data near the groundwater observation wells were also collected to be used as model inputs for prediction of groundwater levels. The datasets included the monthly values of groundwater level (GW), total precipitation (P), total evaporation (ET), and average temperature (T). The measured values of daily P, ET, and T data for the Zhangye and Gaotai stations were obtained from the China Meteorological Data Sharing Service System (http://cdc.cma.gov.cn/home.do) for the period June 2003 to December 2010. As the purpose of this study was to predict groundwater for 1-, 2-, and 3-monthly time steps, the daily P and ET were summed up to produce their total monthly equivalent values, while the daily T was averaged to obtain the monthly average temperature value.

In order to develop the ANN and the WA-ANN models, the data were partitioned into two sets: the first set as the training set and the other as the testing set. Subsequently, the values of 2003/6–2008/12 were assigned for the training and the values of 2009/1–2010/12 for testing of WA-ANN and ANN models. Table 1 shows the hydrological statistics for the training, testing, and the total dataset for monthly groundwater level, precipitation, evaporation, and temperature. According to the statistical properties, no statistically significant differences between the divisions of the data were observed. Obviously, this meant that the training dataset contain sufficient information about the system behavior to qualify as a system model.

Selecting appropriate input variables is important for the WA-ANN and ANN model development since a proper set of predictor variables will provide adequate information about the hydrological system that is being modeled. For groundwater level forecasting, the most frequently used inputs are the antecedent groundwater level and the climatic data (e.g., precipitation, evaporation, and temperature) (Adamowski & Chan 2011) as these variables are intrinsically related to groundwater level. In the present study, different combinations of the monthly groundwater level, total precipitation, total evaporation, and average temperature were used as inputs for the WA-ANN and the ANN models in order to forecast the 1-, 2-, and 3-month ahead groundwater levels. For the case of Well I, the data of Zhangye station were used as inputs, similarly, the data of Gaotai station were used as inputs for Well II.

Thus, the three sets of input combinations evaluated in this study were: (A) ancient groundwater level only; (B) ancient climate data including the monthly total precipitation, total evaporation, and average temperature; and (C) ancient groundwater level and ancient climate data.

where t is the current time, Δt is the lead time period (from 1 to 3 months).

The WA-ANN model was developed by combining DWT of the input parameters with the classical ANN model. DWT decomposed the original groundwater level time series into its cyclic (frequency) components (i.e., yielding the DWCs) at different scales (or frequencies). Each component of DWT had a distinct role in the original groundwater level time series. The low-frequency component reflects the identity (periodicity and trend) of the signal whereas the high-frequency component uncovers details (Kucuk & Ağirali-super 2006). Consequently, the WA-ANN was a classical ANN model that used sub-time series components (Ds) obtained using DWT on the original dataset. The inputs of the current WA-ANN model were the Ds of input combinations and the outputs were assigned as the observed monthly groundwater level with a lead time of 1 month, 2 months, or 3 months in advance of the current month.

Each of the previous groundwater levels of Well I and Well II, and the monthly total precipitation, evaporation, and average temperature of Zhangye and Gaotai stations were decomposed into one approximation and four detailed sub-series using the DWT algorithm. It is imperative to note that choice of the mother wavelet for DWT is very important. Generally, mother wavelets with a compact support form (e.g., Daubechies-1, Haar; and Daubechies-4, db4) are the most effective in generating time localization characteristics for time series which have a short memory and short duration transient features (Nourani et al. 2014). In contrast, mother wavelets with a wide support form (e.g., Daubechies-2, db2) yield reliable forecasts for time series with long-term features (Maheswaran & Khosa 2013). In the present work, as the purpose was to forecast the 1-, 2-, and 3-monthly forecasts, the Daubechies-2 (db2) mother wavelet was utilized. For a monthly time series, the model with four levels’ decomposition performed better (Maheswaran & Khosa 2012); therefore, the detailed components represented the 2-month periodicity (D1), 4-month periodicity (D2), 8-month periodicity (D3), and 16-month periodicity (D4), respectively. The approximation A4 represented the approximation components at the fourth level of decomposition of the predictor datasets. As the detailed sub-series played a distinct role in the original time series, and had different effects on the original groundwater time series, all sub-series of wavelet coefficients were considered equally important as they were expected to contain crucial information about the original time series (Seo et al. 2015). Therefore, new input series was constructed by using all DWT components at the different levels considered for groundwater level forecasting.

where X_norm is the normalized data, X_min is the minimum value of the data, and X_max is the maximum value of the data.

After the input and output variables were selected, the optimum ANN model architecture was investigated for both study stations. This involved the determination of the optimum number of neurons (N) in the hidden layer. As there is no ‘rule of thumb’ to determine the hidden neuron structure, the optimal number of neurons in the hidden layer was identified using a trial and error procedure (Deo & Şahin 2015). This required gradual variation in the number of hidden neurons from 2 to 50, with each architecture configuration randomized 100 times in order to deduce the optimum network with the lowest mean square error.

where n is the number of input samples; and are the observed and predicted groundwater levels at time t, are the mean of the observed and predicted groundwater levels, σ_o denotes the standard deviation of observations. The ideal fit between the observed and predicted values are likely to demonstrate R = 1, MAE = 0, RMSE = 0, NS = 1, and RSR = 0, respectively. However, in accordance with the literature, normally, a predictive model can be considered very good if the NS value is greater than 0.75 and RSR is less than 0.50, good performance if NS is greater than 0.65 and RSR is less than 0.6, and satisfactory performance if NS is greater than 0.5 and RSR is less than 0.70 (Moriasi et al. 2007).

In this section, the optimum WA-ANN model is benchmarked against the classical ANN model, and its predictive skill in simulating the 1-, 2-, and 3-month ahead groundwater level for Well I and Well II are assessed. Tables 2 and 3 show a summary of the results for simulation of groundwater level.

As the purpose of this research is to demonstrate the efficacy of the WA-ANN model for groundwater level forecasting at multiple timescales, the prediction metrics for the WA-ANN model are summarized (Tables 2 and 3). According to simulations of groundwater level obtained in the testing period, it was obvious that as the lead time of forecast increased, the accuracy of WA-ANN models decreased. This result was consistent with other research works modeling water resources on multiple timescales (Nayak et al. 2006; Moosavi et al. 2013), where forecasting skill of the model deteriorated with increase in forecasting timescale, which is perhaps attributable to the reduction in overall data patterns for longer time steps. When using only past groundwater level data as the input (combination A) for 1-month ahead forecasting (Tables 2 and 3), the WA-ANN models achieved very good performance for Well I and Well II on the basis of NS and RSR values. That is to say, this model obtained the R, RMSE, MAE, NS, and RSR statistics of 0.973, 0.086, 0.109, 0.941, and 0.237 for Well I, respectively; and yielded the R, RMSE, MAE, NS, and RSR statistics of 0.959, 0.091, 0.106, 0.913, and 0.287 for Well II, respectively. It can be seen that the R values of both wells were significantly high for this timescale (>0.90), while the MAE and RMSE values were relatively small, thus indicating very good forecasting skill of the model. From the same standpoint, this also verifies a significantly strong correlative relationship between the forecasted and observed groundwater level values and, therefore, indicates the role of memory in present and antecedent values for predicting the future values of the parameter.

For the simulation at the 2- and 3-monthly timescales, a comparative analysis of the statistics in the test period showed that the WA-ANN model performance was worse than the 1-month timescale, with a notable reduction in the magnitude of R and NS, and an increase in the RSR, MAE, and RMSE for both wells. However, the magnitudes of these parameters remained within reasonable accurate range of model performance. Despite the slightly deteriorated performance at longer timescales, we observed that the NS values were greater than 0.80 and the RSR was less than 0.50 for Well I and the NS values were greater than 0.65 and the RSR values were less than 0.60 for Well II. This indicated that the WA-ANN provided very good results for Well I and good results for Well II in 2- and 3-month ahead groundwater level forecasting. Moreover, the WA-ANN model achieved lower RMSE and MAE but higher R for both wells considered in this research. As a result, it is deduced that WA-ANN can yield effective predictions when using only ancient groundwater level data as inputs.

In this section, the forecasting results of groundwater level using ancient total precipitation (P), total evaporation (ET), and average temperature (T) are presented where model simulations are performed via Equation (9). Note that the groundwater level was excluded in this analysis in order to check the overall change in predictive response of the WA-ANN model. According to the performance of the model in the testing periods (Tables 2 and 3), it is clearly shown that the performance of the WA-ANN model appears to remain within the threshold of a very good model for Well I, that is, the value of NS was between 0.822 and 0.900 and RSR remained low (0.307–0.411). For Well II, the NS values for 1-, 2-, and 3-month ahead predicting were 0.670, 0.582, and 0.531, and RSR values were 0.559, 0.631, and 0.669, respectively. It appears that the WA-ANN model can provide good performance forecast for 1-month ahead and satisfactory performance forecast for 2- and 3-month ahead groundwater level. Therefore, in terms of the magnitude of R, RMSE, MAE, NS, and RSR statistics, the WA-ANN model was able to accurately forecast the 1-, 2-, and 3-month ahead groundwater level for both wells using the ancient total precipitation, total evaporation, and average temperature as its input variables. This result is not surprising, taking into account the fact that groundwater levels are directly related to precipitation, evaporation, and temperature, and so, it is justified that precipitation and evaporation behaved as the two main hydrological parameters that directly influence the groundwater recharge and discharge. Likewise, the temperature variable is a good surrogate that reflects potential water uses, for example, warmer temperatures may also increase water consumption, leading to larger groundwater withdrawals since higher temperatures are often correlated with increased water use. In such cases, the temperature can be considered as a groundwater pumping factor (Lian 2011). It is also imperative to mention that from a practical perspective, the WA-ANN that produced relatively accurate results for the 1-, 2-, and 3-month ahead groundwater level by employing only the ancient total precipitation, total evaporation and average temperature is particularly useful for decision-making where reliable hydrogeological information are not available.

In the next stage, the predictive skill of the WA-ANN model was tested by the inclusion of the ancient groundwater levels and meteorological dataset, namely, Equation (10). In this case, the primary purpose was to assess the impact on the model simulations if a larger set of predictor variables including ancient groundwater levels, total precipitation, evaporation, and average temperatures were fed into the model. Tables 2 and 3 show unambiguously, that the inclusion of both the hydrological and climate parameters led to an overall improvement in simulation skill of the WA-ANN model at the 2- and 3-monthly timescales. However, at the 1-monthly timescale, the model simulations were worse than that with ancient groundwater level data inputs, leading to an overall reduction in the magnitude of R, NS and increases in MAE, RMSE, and RSR.

Compared to the three categories of inputs tested for WA-ANN model development, in general, it was observed that the WA-ANN model with Category A as the input yielded the best performance for 1-month ahead groundwater level prediction. This showed that, as expected, the WA-ANN model was quite versatile in extracting the pertinent features embedded in the groundwater level time series and, therefore, produces the most accurate simulation of future trends of groundwater level. However, when only the climate data inputs were used, the model performance deteriorated primarily because the inputs no longer contained the best set of explanatory features as much as the groundwater level time series did. Notwithstanding this, integrating the groundwater level and climate data into the WA-ANN models significantly improved its performance by reducing the RMSE, MAE, and RSR and elevating the values of R and NS for the 2- and 3-month ahead groundwater level forecasting. In other words, it is likely that over the short-term period, groundwater level appeared to be moderated by the groundwater dynamics, while over the long term, groundwater level fluctuations are dependent on groundwater dynamics, climate parameters as well as human interventions within the monitoring sites. Furthermore, the WA-ANN model exhibited different forecasting abilities at each groundwater level observation well. This suggested that the predictability at different sites could be strongly affected by the factors, including the nature of the groundwater level fluctuations, local climate conditions (e.g., rainfall, evaporation, and temperature), and hydrological features in a certain sub-hydrogeological zone. Therefore, the prediction of groundwater at climatologically diverse locations must assess the relevance of the factors impacting this property in order to simulate it with good accuracy.

While statistical parameters presented so far have assessed the overall skill of WA-ANN models in simulating groundwater levels, hydrograph and scatter plots for Well I and Well II were also useful in assessing the temporal correspondence of the observed and model simulated values. Figures 2 and 3 show the observed and the predicted values of groundwater levels at the 1-, 2-, and 3-month lead times within the test period. It was observed visually that the WA-ANN technique performs very well for groundwater level predictions with 1-month lead prediction better than for 2- and 3-month lead period, with the performance for higher lead times deteriorating gradually. Furthermore, it was obvious that the groundwater level values estimated by the WA-ANN models are closely matched with the observed values and followed the same trend in all plots.

For practical applications, especially for real-time forecasting, the level of accuracy is particularly important for reliable groundwater supply planning (Coulibaly et al. 2001). Figures 2 and 3 show that the proposed WA-ANN models exhibited good prediction accuracy for high values of groundwater levels but, in fact, were unable to maintain accuracy for the low values of groundwater levels. For example, when ancient climatic data such as the monthly total precipitation, total evaporation, and average temperature were used as inputs, the lowest groundwater level data obtained in the test period at the 1-, 2-, and 3-month ahead were 1,446.29 m, 1,445.93 m, and 1,446.04 m for Well I, respectively, while the lowest value for Well I was 1,445.73 m. Similarly, the lowest values were considerably underestimated in the test period of the WA-ANN model, 0.16 m, 0.19 m, and 0.31 m in 1-, 2-, and 3-month ahead prediction, respectively. This can be considered an important limitation of this study, as also noted in other studies that used AI models for groundwater prediction (Coulibaly et al. 2001; Sudheer et al. 2011). Despite this, it can be seen that the WA-ANN models provided highly accurate forecasts.

In order to assess the ability of the WA-ANN model relative to the regular ANN model for the 1-, 2-, and 3-month ahead groundwater level forecasting, the latter model was developed using the original data as input categories of (A), (B), and (C) for groundwater level predicting. The best ANN model for groundwater level forecasting of Well I and Well II is shown in Tables 4 and 5. It can be clearly seen that the WA-ANN models performed much better than the ANN models not only for 1-month ahead groundwater level forecasting but also for 2- and 3-month ahead forecasting. Concretely, the WA-ANN model produced a lower MAE, RMSE, RSR as well as higher R and NS values. Using Well I as a representative example, using ancient groundwater level and climatic data as inputs, for the lead time of 1-month ahead forecasting, R value obtained for the ANN model was 0.921, however, for the WA-ANN model, it increased to 0.973. Similarly, the values of MAE, RMSE, NS, and RSR values obtained by the ANN were 0.114, 0.174, 0.867, and 0.382 in contrast to equivalent values of 0.091, 0.117, 0.932, and 0.254 for the WA-ANN model. It is noteworthy that the WA-ANN model improved the values of R, MAE, RMSE, NS, and RSR by 5.65%, 20.18%, 32.76%, 7.49%, and 33.51%, respectively, in comparison to the ANN model. Moreover, compared with Tables 2 and 3, it was also evident that the performance of the WA-ANN deteriorated more slowly than that of the ANN model. As the lead time of forecast increased, the WA-ANN performance improved more than the performance of the ANN model. Clearly, the WA-ANN model was able to provide more accurate forecasting than the regular ANN model, especially for the longer lead time forecasting.

Figures 4 and 5 show the hydrograph and scatter plots of Well I and Well II using the regular ANN model. According to Figures 2 and 3, the simulations of the WA-ANN model were closer to the observed groundwater levels compared to the ANN model. Considering the best fit equation of the form y = ax + b where a and b are the coefficients for the WA-ANN and ANN model, respectively, the WA-ANN model yielded a and b closer to the 1 and 0 compared to the ANN model, indicating significantly better fit. This indicated that the WA-ANN model has a better generalization skill of the predictive data compared to the regular ANN model with no wavelet filtering.

For the lowest groundwater level simulations, the WA-ANN and the ANN models appeared to underestimate the values of groundwater level. However, a significant improvement was observed in most cases for the WA-ANN compared to the ANN model (Figure 6). For example, when taking Category A as an input, we note that the lowest groundwater level values were considerably underestimated in the test period of the WA-ANN simulations, with magnitudes of 0.17 m, 0.42 m, and 0.52 m for Well I, and 0.14 m, 0.33 m, and 0.44 m for Well II in 1-, 2-, and 3-month ahead prediction, respectively. The ANN model underestimated it by 0.37 m, 0.43 m, and 0.73 m for Well I, and 0.47 m, 0.64 m, and 0.63 m for Well II in 1-, 2-, and 3-month ahead prediction. Clearly, WA-ANN model seems to be more suitable for modeling the minimum values of groundwater level.

While assessing the performance of any model for its applicability in predicting groundwater level, it is not only important to evaluate the average prediction error but also the distribution of prediction errors. Figure 7 shows a comparison between the WA-ANN and ANN methods for the degree of dispersion and skewness of error for both wells using a boxplot diagram. One can see that although the ANN model yields small error distribution, the WA-ANN model produces smaller errors and the dimensionless residual is smaller than the ANN model in most cases. This is true not only for 1-month ahead groundwater level forecasting but also for the 2- and 3-month ahead forecasting values. Therefore, the present results show the WA-ANN model to be a superior model compared to the classical ANN based on prediction error distributions.

This study has shown very good efficiency of the WA-ANN model in simulating the groundwater level for the present study location, although the available data records were relatively short. Furthermore, the findings revealed that the WA-ANN model was more adequate than the regular ANN with no filtering of input data, for 1-, 2-, and 3-month ahead groundwater level forecasting. As such, the WA-ANN model can be employed successfully in groundwater level estimation in a complex groundwater system with an arid environment. Another benefit of the WA-ANN model, which required only the ancient monthly total precipitation, evaporation, and average temperature as inputs to yield reasonably accurate simulations, is that it can be considered as a simple, yet robust model with significant potential for prediction of monthly groundwater level in a complex and dynamical aquifer system.

Although our results are promising in regards to the application of the WA-ANN model for forecasting groundwater level in complex groundwater systems, it has some limitations. First, in our work, the WA-ANN model generally underestimated the corresponding lowest groundwater levels and thus was unable to predict effectively the lowest records of groundwater in some of the cases. Therefore, further research is necessary to improve its prediction accuracy, especially for the lower values of groundwater level, by combining or improving the predictive model parameters. Second, our model did not consider the river level as input. In an inland river basin, the hydrodynamics of water exchange between the groundwater and river flows are frequent. Therefore, if this factor is taken into account, more accurate predictions of groundwater levels may be possible. Third, in this work, we focused on the performance of the WA-ANN model for 1-, 2-, and 3-month ahead groundwater level forecasting; however, hybrid models combining wavelet with other AI models may also be interesting to further explore their potentials of both short- and long-term forecasting.

In arid environments, groundwater plays an important role as a water supply both for industry, drinking, agriculture, and ecosystems. Therefore, accurate and reliable groundwater level forecasting models are important for the management of groundwater resources. In this paper, the potential of the WA-ANN model for 1-, 2-, and 3-month ahead groundwater level forecasting was investigated in a complex groundwater system of arid inland river basin, northwestern China. The WA-ANN model was developed and tested by employing three different input combinations, including the ancient groundwater levels, the ancient meteorological data such as the monthly precipitation, evaporation, and average temperatures, and ancient groundwater level combined with the ancient climatic data. The original input data were decomposed into the sub-series by DWT. All sub-series coefficients were treated as equally important and used as input for the WA-ANN model. Based on our results, we found that the WA-ANN model can provide accurate forecasts for 1-, 2-, and 3-month ahead groundwater level fluctuations of all categories of inputs. The results showed that the WA-ANN model with ancient groundwater level as the input variable yielded the highest performance for the 1-month ahead groundwater level prediction. However, the WA-ANN models that integrated ancient groundwater level and ancient meteorological data performed the best for 2- and 3-month ahead groundwater level forecasting. Importantly, the WA-ANN model with only the ancient monthly total precipitation, total evaporation, and average temperature also showed accurate results for 1-, 2-, and 3-month ahead groundwater level fluctuations, although its performance was slightly worse than the model with both groundwater and climate data as inputs. Thus, based on our results, the WA-ANN model can be considered as a simple, yet robust model that offers significant potential for accurate prediction of monthly groundwater level by a lack of appropriate hydrogeological information.

Based on the comparison of the WA-ANN performances with the regular ANN models for 1-, 2-, and 3-month ahead groundwater level forecasting, the WA-ANN models provided more accurate results than the regular ANN models. Also, the WA-ANN model was a useful new tool for the long-term groundwater level forecast problem, since the wavelet transform tends to enhance the ANN's forecasting ability. Finally, we propose that the WA-ANN model can be particularly relevant for forecasting the complex dynamical changes in groundwater level fluctuations with good accuracy. Such models can be useful for decision-making without the need for underlying physical relationships between groundwater and its interacting properties, especially in aquifers when there are limited input data or hydrogeological information is unavailable.

This work was funded by the Key Research Program of Frontier Sciences, CAS (QYZDJ-SSW-DQC031), National Natural Sciences Foundation of China (31370466) and National Key Research and Development Program of China (2016YFC0400908). All authors also thank anonymous reviewers for their reading of the manuscript and for their suggestions and critical comments.