Estimating Groundwater Levels Using System Identification Models In

Physics and Chemistry of the Earth xxx (2017) 1e7

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Physics and Chemistry of the Earth

journal homepage: www.elsevier.com/locate/pce

Estimating groundwater levels using system identiﬁcation models in Nzhelele and Luvuvhu areas, Limpopo Province, South Africa

* Rachel Makungo , John O. Odiyo

Department of Hydrology and Water Resources, University of Venda, Private Bag X 5050, Thohoyandou, 0950, South Africa article info abstract

Article history: This study was focused on testing the ability of a coupled linear and non-linear system identification Received 2 June 2016 model in estimating groundwater levels. System identification provides an alternative approach for Received in revised form estimating groundwater levels in areas that lack data required by physically-based models. It also 20 October 2016 overcomes the limitations of physically-based models due to approximations, assumptions and simpli- Accepted 31 January 2017 fications. Daily groundwater levels for 4 boreholes, rainfall and evaporation data covering the period Available online xxx 2005e2014 were used in the study. Seventy and thirty percent of the data were used to calibrate and validate the model, respectively. Correlation coefficient (R), coefficient of determination (R2), root mean Keywords: fi fi fi Coupled linear and non-linear square error (RMSE), percent bias (PBIAS), Nash Sutcliffe coef cient of ef ciency (NSE) and graphical ts 2 Groundwater levels were used to evaluate the model performance. Values for R, R , RMSE, PBIAS and NSE ranged from 0.8 to Model performance 0.99, 0.63 to 0.99, 0.01e2.06 m, 7.18 to 1.16 and 0.68 to 0.99, respectively. Comparisons of observed and System identification model simulated groundwater levels for calibration and validation runs showed close agreements. The model performance mostly varied from satisfactory, good, very good and excellent. Thus, the model is able to estimate groundwater levels. The calibrated models can reasonably capture description between input and output variables and can, thus be used to estimate long term groundwater levels. © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction systems is generally carried out by means of physically-based models, which demand a proper synthesis of the aquifer parame- Groundwater is a strategic water resource that aids in supplying ters to describe the spatial variability of the subsurface (Taormina domestic needs in most developing countries. Groundwater levels et al., 2012). Data required to quantify aquifer parameters is from observation wells provide a principal source of information rarely available and expensive to acquire in most developing regarding the hydrological stresses acting over aquifers and how countries. In addition, approximations, assumptions and simplifi- those stresses influence groundwater recharge, storage and cations that are made in physically-based models result to errors discharge (Sujay and Paresh, 2015). Groundwater levels are and uncertainty in the outputs. System identification models can required for groundwater resource assessment to ensure its sus- overcome some of the limitations of physically-based models in tainable utilisation. However, most boreholes that are drilled in cases where data on aquifer parameters, which is required for most developing countries are typically production boreholes estimating groundwater levels, is not available. aimed at domestic water supply. This in addition to poor ground- System identification is the art and science of building mathe- water monitoring networks and discontinuous monitoring or matical models of dynamic systems from observed input-output measurement of groundwater levels results in lack of continuous data (Ljung, 2010). In system identification or time series analysis, long term groundwater levels time series data. This creates the the groundwater system is seen as a black box that transforms a need to estimate and extend limited groundwater levels data. series of observations of the input or explanatory variables into a Simulation of hydraulic heads fluctuations in groundwater series of the output variables or groundwater levels (von Asmuth and Knotters, 2004). Using a time series model, it is possible to simulate periods without observations, as long as data on explanatory series are available (Manzione et al., 2009). Different model * Corresponding author. classes that are used in system identification include linear, non- E-mail addresses: [email protected] (R. Makungo), john.odiyo@ linear, hybrid, discrete, continuous, non-parametric, amongst univen.ac.za (J.O. Odiyo). http://dx.doi.org/10.1016/j.pce.2017.01.019 1474-7065/© 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article in press as: Makungo, R., Odiyo, J.O., Estimating groundwater levels using system identification models in Nzhelele and Luvuvhu areas, Limpopo Province, South Africa, Physics and Chemistry of the Earth (2017), http://dx.doi.org/10.1016/j.pce.2017.01.019 2 R. Makungo, J.O. Odiyo / Physics and Chemistry of the Earth xxx (2017) 1e7 others. Description of these models can be found in existing liter- ature including Ljung (2014). There are several applications of system identification models in groundwater levels estimation studies. von Asmuth and Knotters Fig. 2. Structure of NLHW model. (2004) used a method based on continuous time transfer function (TFN) model, which estimates the impulse response function of the system from the temporal correlation between time series of yðtÞ¼hðxðtÞÞ (2) groundwater level and precipitation surplus, to describe groundwater dynamics. Bierkens et al. (2010) modelled the spatiotemporal variation of shallow water table depth using a regionalised version of an autoregressive exogenous (ARX) time series. von Asmuth 3. Study area (2012) developed Menyanthes software in which groundwater levels time series can be modelled using both the ARMA and PIR- The study area covers selected quaternary catchments in FICT methods. The current study used a coupled linear polynomial Nzhelele (A8) and Luvuvhu (A9) River Catchments (Fig. 3) in Lim- Output-Error (OE) and non-linear Hammerstein-Wiener (NLHW) popo Province of South Africa. Mean annual precipitation (MAP) for system identification model for estimating groundwater levels. This Luvuvhu River Catchment varies from 1800 mm in the moun- system identification approach has not yet been tested for tainous areas located in the north-western part to 300 mm at the groundwater levels estimation. Thus, the study is aimed at testing eastern side of the catchment where it joins the Limpopo River the ability of coupled linear polynomial OE and NLHW model in (Kagoda and Ndiritu, 2009). The MAP for Nzhelele River Catchment estimating groundwater levels. varies from 350 to 400 mm. The study area falls within the severely faulted Soutpansberg Group of the Mokolian age. The Soutpansberg Group forms part of fractured crystalline basement aquifers of Limpopo Province in 2. Structure of the model used in the study South Africa. Dykes and sills of diabase are plentiful in the Sout- pansberg rocks (Brandl, 2003). Crystalline basement rocks are fi The model used in the current study consists of coupling linear usually semi-con ned (fractured bedrock) with water-table aqui- polynomial OE and NLHW models. Linear polynomial OE model fers (the matrix-regolith) situated on top of them (Holland, 2011). structure (Fig. 1) is one of the various linear model structures that The study area is characterized by fractured aquifers with median e provide different ways of parameterising the transfer functions of borehole yield of 0.5 2 l/s (du Toit et al., 2002). linear input-output polynomial model within system identification software (Ljung, 2014). 4. Methodology

BðqÞ Daily groundwater levels from 4 boreholes, rainfall and evapo- yðtÞ ¼ uðt n Þ þ eðtÞ (1) ration data (Table 1) from selected quaternary catchments of FðqÞ k Nzhelele and Luvuvhu Catchments were used in the study. Data covered the period 2005e2013. Knotters and van Walsum (1997) ð Þ : ¼ þ 1 þ ::: þ nbþ1 e B q nb b1 b2q bnbq used data with limited lengths (4 10 years) to develop time series models that enabled simulation of water table depths of extensive length (30 years). Rainfall and evaporation data within ð Þ : ¼ þ 1 þ ::: þ nf F q nf 1 f1q fnf q the vicinity of each borehole were inputs into the model. Using data within the vicinity of each borehole overcomes the effects of spatial nb and nf are orders of the polynomials B and F, respectively, and nk climate variability (Knotters and van Walsum, 1997). Rainfall and is the delay from input to output in terms of number of samples, evaporation are the explanatory variables of the model. Manzione 1 q is time-shift operator, u is the input, y(t) is the output at time (t) et al. (2009) also incorporated precipitation and evapotranspira- and e(t) is model error. NLHW model (Fig. 2) represents dynamics tion as exogenous variables into the model when mapping water of a system by a linear transfer function and captures the non- table depths since they are the most important driving forces of linearities using nonlinear functions of inputs and outputs. NLHW water table fluctuations. fi model achieves this con guration as a series connection of static A coupled linear polynomial OE and NLHW model was used to non-linear blocks with a dynamic linear block. Detailed description estimate groundwater levels in each borehole, within the System of OE and NLHW models is provided in Ljung (1998, 2014). Identification Toolbox of MATLAB. Wavelet network (Equation (3)) ¼ w(t) f(u(t)) is a nonlinear function transforming input data, was selected as nonlinearity estimator. u(t) at time, t. x(t) ¼ (B/F)w(t) is a linear transfer function. B and F are similar to polynomials in the linear polynomials OE model Xn ð Þ¼ ð ð ÞÞ (Equation (1)), and f and h are scalar functions for input and output g x akk bk x gk (3) channels, respectively. The model output, y(t) is computed by: k¼1

g(x) is wavelet network, bk is a row vector such that bk (xgk)isa scalar. The linear polynomial OE model was used to initialise the NLHW model. The initialisation configures the NLHW model to use orders and delays of the linear model, and polynomials as the transfer functions (Ljung, 2014). This initialisation aids in improving the fit of the model. Seventy and thirty percent of the data were used to calibrate and validate the model, respectively. Model performance was evaluated using graphical fits, correlation coefficient (R), co- 2 Fig. 1. Structure of linear polynimial OE model. efficient of determination (R ), root mean square error (RMSE),

Fig. 3. Study area.

Table 1 Table 2 Stations used and study periods. Model performance evaluation criteria.

Borehole Rainfall Evaporation Period R2 >0.85 Excellenta 0.75e0.85 Very gooda A9N0009 (Tshidzivhe) 0766827W None 2007/10/08e2013/11/14 0.5e0.65 Satisfactorya A9N0018 (Luvuvhu) 0723485W A9E002 2005/07/02e2012/11/25 R >0.9 Satisfactoryb A8N0515 (Maangani) 0766201W A8E004 2006/11/16e2013/11/08 0.5 Acceptablec A8N0508 (Mandala) 0766563W A8E004 2006/02/03e2012/03/25 RMSE (m) 0 Perfectd NSE 0.9 Very goode 0.8e0.9 Goode e e Nash Sutcliffe coefficient of efficiency (NSE) and percent bias 0.65 0.80 Acceptable PBIAS (%) <10 Goodc (PBIAS). A combination of graphical results, error statistics (RMSE a and PBIAS), and goodness-of-fit statistics (R, R2, and NSE) is Yan et al. (2014). b fi Singh et al. (2004). essential to ensure accurate veri cation of the model (see Ritter and c ~ van Liew et al. (2007). Munoz-Carpena, 2013). Table 2 shows the model performance d Shamsudin and Hashim (2002). evaluation criteria used in the current study. e Ritter and Munoz-Carpena~ (2013).

5. Results and discussion were used in the modelling. For A9N0009, single model order was used since one input variable (rainfall) was used in the modelling. Table 3 shows selected model orders which gave the best results Rainfall was used as the only input in the model for A9N0009 since after several trial runs. The model orders define the number of there was no evaporation station within its vicinity. terms in each equation. Model calibration resulted to equations for Fig. 4 shows the estimated and observed groundwater levels for B(q) and F(q) polynomials defined in Equation (1) and their co- boreholes A9N0009, A8N0508, A8N0515 and A9N0018 for both efficients. Thus, the model orders and coefficients of the equations calibration and validation runs. The comparisons of observed and are unique parameters obtained from the calibration process. The estimated groundwater levels for calibration and validation runs model orders and equations for A9N0018, A8N0515 and A8N0508 mostly show close agreement between observed and simulated are in pairs since two input variables (rainfall and evaporation)

Table 3 Model orders used in groundwater modelling.

Borehole Calibrated nb, nf, nk Polynomial B(q) Polynomial F(q) Model error (e(t))

2 3 1 A9N0009 2, 1, 2 Bðq1Þ0:312q 0:357q Fðq1Þ¼1 0:999q 0.0338 3 1 1 1 A9N0018 2,2; 2,2; 1,1 Bðq1Þ¼6:457e 05q Bðq2Þ¼0:008q Fðq1Þ¼1 0:999q Fðq2Þ¼1 0:999q 1.5836 3 4 5 6 1 2 3 4 A8N0515 4,4; 4,4; 3,3 Bðq1Þ¼0:217q 0:111q 0:168q þ 0:131q Fðq1Þ¼1 0:438q 1:376q þ 0:284q þ 0:531q 0.0113 3 4 5 6 1 2 3 4 Bðq2Þ¼0:029q 0:014q 0:067q þ 0:512q Fðq2Þ¼1 1:64q 0:0650q þ 0:842q þ 0:267q 4 5 6 7 1 2 3 A8N0508 4,4; 3,3; 4,4 Bðq1Þ¼0:025q 0:029q 0:029q þ 0:025q Fðq1Þ¼1 1:602q 1:06q þ 0:4568q 0.1359 4 5 6 7 1 2 3 Bðq2Þ¼0:0004q 0:006q 0:006q þ 0:004q Fðq2Þ¼1 1:444q 0:05654q þ 0:5003q groundwater levels. The shapes of the observed and estimated RMSE values obtained in the current study are comparable to those groundwater levels graphs are also similar. General agreement of related studies with exception of those for A9N0009 and vali- between observed and simulated frequencies for the desired con- dation run of A9N0018, which were higher. In general, the perfor- stituent indicates adequate simulation over the range of the con- mance of borehole A9N0009 was not consistently good as those of ditions examined (Singh et al., 2004). The model underestimated the three boreholes that had two input variables. This indicates that peak groundwater levels for calibration and validation runs for the use of more input variables results in better performance. The A9N0009 (Fig. 4). Peak groundwater levels which occurred on graphical fits and measures of performance generally show efficient 2010/10/28 and 2013/01/23 in the calibration run of A8N0515 were calibration and validation of the model for each borehole. Thus, not well estimated. Groundwater levels for validation runs of rainfall and evapotranspiration can be used to simulate ground- A9N0018 were mostly slightly underestimated. In borehole water levels based on the coupled linear polynomial OE and NLHW A8N0508, noticeable underestimation of groundwater levels system identification model. within the period 2011/10/21 to 2011/12/28 and overestimation Fig. 5 shows general increases and decreases in groundwater within the period 2011/12/30 to 2012/02/12 occurred in the vali- levels which correspond to increases and decreases in rainfall, dation run. Groundwater levels were also underestimated in the respectively. Results from multi-site monitoring in a study by van validation run of A8N0515 in the periods 2007/09/09 to 2007/09/19 Wyk et al. (2012) emphasised that a direct recharge mechanism, and 2008/02/04 and 2008/02/26. Black box models do not capture which is enhanced by the presence of macro-pore features, exists in the physical processes particularly those that occur during extreme fractured hard-rock terrains of South Africa. This was indicated by rainfall events and thus, they are likely to underestimate peak the short lag-time (1 he5 days) between rainfall events and water groundwater levels. table responses (van Wyk et al., 2012). Since, the study area falls Values of R2 for three of the boreholes (A9N0018, A8N0515 and within the fractured hard-rock terrains of South Africa, the lag time A8N0508) including calibration run of A9N0009 were greater than between rainfall and groundwater levels is also expected to be 0.85 showing excellent model performances (Table 4). R2 value for short. This was accounted for by e(t) component of the model. The validation run of A9N009 fell in the range of 0.5e0.65 showing e(t) in Equation (1) accounts for the unknown physical processes satisfactory model performance. von Asmuth and Knotters (2004) that influence groundwater levels. These include the influence of obtained R2 values of 81.5 and 91.9% on calibrating continuous hydrogeologic environment where groundwater recharge, storage time TFN system identification model for characterising ground- and flow occurs, groundwater abstractions, and lag time between water dynamics using groundwater levels data. These values are rainfall and groundwater levels in the study area. In studies by mostly comparable to those obtained in the current study and Knotters and van Walsum (1997) and Manzione et al. (2009), the indicate very good to excellent performances. noise component of the TFN models were used to describe part of NSE values for calibration and validation runs for A9N0018 and the water table behaviour that could not be explained from the A8N0515, and calibration run for A8N0508 were greater than 0.9 used physical concepts or empirically from the input series. (Table 3), showing that their models performances were very good. The macro-pores from severely faulted crystalline basement Calibration and validation runs for A9N0009 and A8N0508, aquifers and their interconnectivity result in good response of respectively, had NSE values within the range of 0.8e0.9, showing groundwater levels to rainfall events in the study area and hence a good model performance. High NSE values indicate less variance good relationship between the two. Thus, rainfall is one of the error (van Liew et al., 2003). All R values were >0.9 except for the major drivers of groundwater levels fluctuations in the study area. validation run of A9N0009, thus generally showing satisfactory In addition, the good response of groundwater levels to rainfall model performance. A9N0009 had R value of 0.80, which exceeded events explains good model performance based on graphical fits the acceptable value of 0.5. and performance measures for majority of the boreholes. A study PBIAS values for all boreholes were less than 10% showing good by Ochoa and Reinoso (1997) reported that increments in the water model performance. RMSE values for A9N0018, A8N0515 and table level could be explained by local precipitation suggesting that A8N0508 for calibration and validation runs were closer to zero, the Donana~ National Park dune aquifer, in the southwestern coast except for validation run of A9N0018. RMSE values close to zero of Spain is very sensitive to local meteorological conditions. indicate perfect fit(Singh et al., 2004). A9N0009 had the highest Hodgson (1978) reported that a comparison of actual and esti- RMSE values of 2.06 and 1.00 m for calibration and validation runs, mated water table responses using linear regression, in the Vryburg respectively. This is most likely because peak groundwater levels area of South Africa, indicated that water table response could be for this borehole were underestimated resulting to more estima- simulated by merely considering rainfall and discharge (ground- tion errors. Average RMSE value of 22 cm was obtained in a study by water abstractions). Hodgson (1978) also noted that improvement Knotters and Bierkens (2001) on predicting water table depths in the model could be achieved by considering other variables of using regionalised autoregressive exogenous variable model. the groundwater balance. Thus, in borehole A9N0009 with high Knotters and Bierkens (2000) obtained RMSE values ranging from RMSE and lower R2, R and NSE during validation compared to other 17.77 to 23.37 cm on validating an autoregressive exogenous vari- boreholes, for example, other variables such as groundwater ab- able model for predicting water table depths. von Asmuth and stractions may aid in improving performance measures if they are Knotters (2004) obtained RMSE values of 8.4 and 11.2 cm. Most considered as input into the model. However, monitored

Fig. 4. Observed and estimated groundwater levels for A9N0018, A8N0515, A9N0009 and A8N0508 for calibration and validation runs.

Table 4 6. Conclusion Computed measures of performance.

Performance Measure A9N0018 A8N0515 A9N0009 A8N0508 This study tested the ability of coupled linear polynomial OE

Cal Val Cal Val Cal Val Cal5 Val and NLHW model in estimating groundwater levels for 4 boreholes in Nzhelele and Luvuvhu Catchments. Comparisons of R2 0.99 0.96 0.97 0.89 0.88 0.63 0.99 0.86 observed and estimated groundwater levels for calibration and R 0.99 0.95 0.98 0.94 0.94 0.80 0.97 0.93 RMSE (m) 0.08 0.47 0.04 0.06 2.06 1.00 0.03 0.01 validation runs mostly showed close agreements, though there NSE 0.99 0.99 0.98 0.98 0.86 0.68 0.99 0.84 were underestimation and overestimation of groundwater levels PBIAS (%) 0.01 1.16 0.02 0.27 0.003 7.18 0.08 0.11 in some boreholes. The model performance mostly varied from

Fig. 5. Relationship between rainfall and groundwater levels. groundwater abstractions in production boreholes within the vi- satisfactory, good, very good and excellent. The model results cinity of groundwater levels monitoring boreholes in the study area show better performance with more input variables. The results of are not available. the study show that the model is able to estimate groundwater Knotters and van Walsum (1997) noted that use of models to levels based on the graphical ﬁts and model performance mea- estimate ﬂuctuating quantities will be successful only if they sures. In addition, the calibration period used in this study is adequately describe the relation between input and output series. adequate to capture reasonable description between input and This implies that the calibration period must be long enough in output variables and can, thus be used to estimate long term order to identify appropriate models and to estimate parameters groundwater levels. accurately. In their study, analyses for two observation wells indicated that a 4-year calibration period contained all information needed to provide a satisfactory description of the relation between Acknowledgements precipitation excess and water-table depth, for semi-monthly time steps and for shallow water-table depths. In the current study, a Department of Water Sanitation and South African Weather minimum period of 4 years was also used in model calibration. Services are duly acknowledged for providing evaporation and Thus, this was assumed to contain all information needed to pro- groundwater levels, and rainfall data used in this study, vide satisfactory description between input and output variables. respectively.

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