Stochastic simulation of groundwater dynamics based on grey theory and seasonal decomposition model in a coastal aquifer of South China

Groundwater is one of the major sources of supply for domestic, industrial and agricultural purposes. It serves as a dependable supply source, especially in some countries where there is a lack of surface water. However, the irrational utilization of groundwater not only exacerbates the contradiction between supply and demand, but also causes many ecological, environmental and geological problems. These problems, caused by overexploitation of groundwater and falling water tables, have brought an extremely disadvantageous influence to bear on human life and social economic development, such as desertification, land salinization, ground subsidence, salt-water intrusion, land crack, mining area geological disasters and so on (Dong 2009). Therefore, in order to effectively manage groundwater, it is important to model and predict fluctuations in groundwater levels. Predicting the potential change in groundwater levels is critical for sustainable management of water resources (Emamgholizadeh et al. 2014) and, in particular, is of the utmost importance for both present and future generations (Mohanty et al. 2010).

Groundwater level, as a dynamic response of external stresses (precipitation, extraction, evaporation, climate and so on), can be considered as an output of the groundwater system, and characterized by trend, periodicity, dependability and randomness. The traditional deterministic methods usually are not capable of solving such problems; subsequently, stochastic time series theory has been explored to solve hydrological problems (Bras & Rodriguez-Iturbe 1985; Lin & Lee 1992; Brockwell & Davis 2010; Doglioni et al. 2010). There has been reported various stochastic mathematical models in groundwater dynamic forecasting, for instance, time series analysis, fuzzy mathematics method, radial basis function network model, artificial neural network, regression analysis, grey Markov chain model, grey system method and so on (Salas 1993; Yeh et al. 1995; Sudheer & Jain 2003;Yeh & Chen 2004; Nayak et al. 2006; Sreenivasulu & Deka 2011; Zhu & Hao 2013; Shirmohammadi et al. 2013; He et al. 2014; Emamgholizadeh et al. 2014). A time series model is an empirical model for stochastically simulating and forecasting the behaviour of uncertain hydrologic systems (Kim et al. 2005; Adhikary et al. 2012; Li et al. 2014). Various stochastic time series models, such as the Markov, Box–Jenkins (BJ) seasonal auto-regressive integrated moving average (ARIMA), depersonalized auto-regressive moving average (ARMA), periodic auto-regressive (PAR), transfer function noise (TFN) and periodic transfer function noise (PTFN), have been applied for these purposes (Mirzavand & Ghazavi 2015). Time series modelling is a useful tool for detecting trends, developing hydrologic or climatic models, forecasting of hydrologic time series and prediction of future climate scenarios (Coulibaly et al. 2001). The application of time series analysis in forecasting groundwater level assumes groundwater system as black or grey box, extracting inherent information by analysing long-term observation series, without the necessity of obtaining other hydrologic parameters, and thus provides the convenience of groundwater dynamic forecasting on a large scale. Most of the research articles (Li & Zhang 2007; Zhou et al. 2007; Yang et al. 2009; Liang 2011; Lu et al. 2014) decompose the groundwater level time series into trend, periodicity and random components to study their characteristics, then combine the three together as an additive or multiplicative model to forecast groundwater levels. Among the numerical techniques, the grey numerical method has become very popular and is recognized as a powerful numerical tool. The theory of the grey system was established during the 1980s as a method for making quantitative predictions. As far as information is concerned, the systems which lack information, such as structure message, operation mechanism and behaviour document, are referred to as grey systems, where ‘grey’ means poor, incomplete, uncertain, etc. It has received increasing application in the field of hydrology (Xu et al. 2008). There are several models for grey theory, and among them, the GM (1,1) method is relatively simple, but can provide a high precision of prediction.

It is known that there is no an universal model that can be applied to any given case in making a simulation of groundwater dynamics. Any mathematical model has its advantages and disadvantages, with different simulation accuracy for a certain observed time series, so it is essential to discover the suitability of a model in a given case. Also, there has been no reporting on the comparative study of decomposition models with the grey method. In order to obtain the suitability of different simulation methods for a given case, in this paper the GM (1,1) method and seasonal decomposition method, multiplicative and additive methods, have been applied to simulate groundwater water tables in a coastal aquifer at Fujian Province, South China, and the simulated results are compared by evaluating root mean square error (RMSE) and regression coefficient (R²).

Time series analysis has been widely used in groundwater resource evaluation, forecast and management due to its simplicity. Multiplicative and additive models are two common models of the seasonal decomposition method, used to analyse the groundwater level data. The model equations can be expressed as follows:

Multiplicative model:

Additive model:

where Y represents the observed groundwater level series, T stands for smoothed trend-cycle (abbreviated as stc), affected by long-term trend factors, S indicates season factors component (abbreviated as saf). I is irregular component (abbreviated as err). The irregular change is a contingency, randomness, burst factor. Specific steps are as follows:

1. Enter the groundwater level data of an observed time series and define the time variable.

2. Draw scatter plot based on the above data, observe the sequence diagram and determine whether it is a stationary sequence.

3. Analyse and process the data. Click the analyse menu bar, select the prediction, and then select the seasonal decomposition of the prediction menu bar, select the multiplicative model or additive model.

4. Explain and illustrate the output results. The results obtained four new additional variables, which are the irregular component (err_1), the sequence after adjustment of the seasonal (SAS_1), seasonal factor (saf_1), and trend cycle component without seasonal and irregular change (stc_1).

5. Draw scatter plot of the four variables mentioned above. According to the chart stc of long-term trends, we can build a linear regression model with polynomial to obtain the relationship between stc and t. Make use of this expression to predict the forecasting value of future trend cycle components stc.

6. According to the 12 values of seasonal factor saf of the multiplicative model, we can get the forecasting value of groundwater level via:

   

   1

   However, the additive model is affected by stc and err, and the predicted groundwater level can be obtained with:

   

   2

   where Z is predictive value of groundwater level, stc is the projections of trend cycle component, saf represents seasonal factors and err denotes the irregular component.

Then the accuracy for stc prediction can be examined by regression coefficient (R²) due to the value in the forecasting process that will produce a certain deviation. Therefore, during the fitting process where the R² value is bigger, the fitting is better. R² value closer to 1 indicates a perfect fitting between the observed and simulated values.

Grey system theory is a multidisciplinary theory dealing with those systems' lack of information, which uses a black-grey-white colour spectrum to describe a complex system whose characteristics are only partially known or known with uncertainty (Deng 1982, 1989). The dynamic of groundwater level is regarded as a typical grey system problem, where the grey GM (1,1) model can better reflect the changing features of groundwater level (Yang et al. 2011). It especially has the unique function of analysing and modelling for short time series, less statistical data and incomplete information of the system, and has been widely applied. GM (1,1) represents an order, a variable, and is a special case of the N variables of differential equation GM (1, N). The essence of GM (1,1) is to accumulate the original data in order to obtain regular data. By setting up the differential equation model, we obtain the fitted curve in order to predict the system. The modelling process is as follows: First, observed data are converted into new data series by a preliminary transformation called AGO (accumulated generating operation). Then, a GM model based on the generated sequence is established. Subsequently, the prediction values are obtained by returning the AGO's level to the original level using IAGO (inverse accumulated generating operation) (Xu et al. 2008).

The specific steps of the GM (1,1) model can be summarized as follows:

1. Suppose the observed original data as

   

   3
2. Accumulate the observed data above once to generate a new sequence, that is

   

   4

   where,
3. According to the newly generated sequence, the differential equation is established as follows:

   

   5
4. Suppose ⁠, a and u can be obtained by least squares estimation

   

   6

   in which,

   

   7
5. Let ⁠, and we get the GM (1,1) forecasting model:

   

   8
6. To obtain the prediction values by returning an AGO's level to the original level using IAGO

   

   9

In this paper, before forecasting the groundwater level, the after-test residue method (Chen et al. 1994) should be used to test the accuracy of the two methods and to find whether these models are suitable for predicting. The two methods mentioned above can be used to calculate the predicted value when the parameters P and C are within the acceptable range, otherwise it is necessary to re-adjust the parameters. The evaluation criteria for the prediction precision are shown in Table 1.

To analyse and forecast the groundwater table in Dongshan County with the two methods mentioned above, taking Kangmei observation well (3506260008) and Chencheng observation well (3506262024) as examples, in which monthly average groundwater level was monitored during 2006–2011, the data set from 2006–2010 is used for model establishment and that of 2011 is used for predicting the dynamic change of groundwater level.

R² measures the degree of correlation among the observed and predicted values. R² values range from 0 to 1. The coefficient of determination describes the proportion of the total variance in the observed data that can be explained by the model. RMSE evaluates the residual between observed and forecasting value. The nearer RMSE value is to 0, the more accurate the forecasting is. The best fit between observed and calculated values will be obtained with R² as 1 and RMSE as 0. It can be seen in Figure 5 that the GM (1,1) model has good prediction effect.

From the accuracy degree of the forecast models of Table 2, it can be seen that that a higher R² value and a lower RMSE of GM (1,1) was obtained for the two selected wells. Therefore, it is concluded that the GM (1,1) model fits the hydrogeological conditions better than the seasonal decomposition model in this case study. On the other hand, the comparison between the two wells indicates that Chencheng observation well had a lower RMSE and higher R², which can be explained by its small fluctuations of groundwater table depth (Figure 2). The method is simple, so the model can be practical. However, if the original data sequence has large fluctuations, the model should be optimized using three point smoothing of the residual model of predicted value and the actual value to obtain a better prediction. In the GM (1,1) prediction model, parameters a and b are fixed once determined, and regardless of the numbers of values, parameters will not change with time. The feature limiting GM (1,1) is that it is only suitable for short-term forecasts because many factors will enter into the system with the development of the system with time. The accuracy of the prediction model will become increasingly weak with the time away from the origin, and thus the predictable significance will diminish.

Groundwater is an important part of water resources, and is also the main source of agricultural irrigation, industrial and domestic water. It is very important to monitor groundwater levels and follow up its dynamic changes. Hence it is essential to obtain a proper method to forecast the groundwater dynamics. Owing to the variety and uncertainty of affecting factors on groundwater levels, stochastic models have been widely applied to groundwater level simulation. However, usually a simple model cannot provide enough accuracy while conducting long-term forecasting, therefore it is necessary to compare different methods in simulating and forecasting an observed time series for a given case. This paper compares the seasonal decomposition method of time series analysis and the GM (1,1) method to simulate and predict groundwater levels in Fujian Province, South China. The effectiveness and the final prediction accuracy of the methods are evaluated with root mean square error (RMSE) and regression coefficient (R²). It is observed that the GM (1,1) model has a higher accuracy in this case. In the case of relatively small fluctuations in the raw data series, the prediction accuracy is high using a grey forecasting model. Application of grey theory to prediction of groundwater level is a novel research area. The model is proposed by virtue of the dynamic characteristics of groundwater level, which increased the forecast precision. Therefore, the method is reliable and effective. However, it is recommended that many methods should be employed to test the reliability and objectivity of the selected model when considering the uncertainties that affect groundwater level dynamics.

This research was financially supported by National Natural Science Foundation of China (grant number 41402202), Specialized Research Fund for the Doctoral Program of Higher Education (20130061120084). Special gratitude is offered to Rita Henderson for her continuous efforts in dealing with and evaluating the work, and the valuable comments from the other two anonymous reviewers are also greatly acknowledged.