Introduction

Understanding and modeling hydrological behavior is an important challenge for managing the water resources and for the investigation of extreme hydrological events, such as floods and droughts. However, a substantial topic in many areas is that meteorological and hydrological data accessibility which is frequently rare. Several problems associated with obtaining reliable long-term hydrological data in semi-arid regions include limited economic resources for monitoring and climates complexity (Beaumont et al. 2016; Wheater 2002). The application of accurate modeling procedures to predict and simulate water availability is an indispensable way for flood protection, irrigation, drought management and water needs. According to Refsgaard and Knudsen (1996) and Wagener et al. (2004), two major categories of rainfall-runoff models can be notable—deterministic models and stochastic models—the first family can be classified into black box models, such as artificial neural networks ANN and statistical models such as Box-Jenkins ARIMA model and conceptual or physically based models such as the GR family. Rainfall-runoff models differ generally in underlying design philosophy. The prior contains forecasting applications (flood warning and hydropower operation) and catchment management purpose (climate impact studies and water distribution); the conclusions are geared on the way to understanding the catchment function and how the individual processes combine to produce catchment response (Parajka et al. 2007). The choice of suitable hydrological modeling for a particular catchment with a special hydrological environment and a distinct physical background should be established on a diversity of aspects containing data availability, model complexity, and prediction uncertainty among others. Consequently, for making a decision on model selection a comparison between the pros and cons of a variety of hydrological models is necessary. In the literature, several studies elaborate the subject of comparative analysis of different hydrological models (Bai et al. 2015; bin Shaari et al. 2017; Machekposhti et al. 2018; McCuen 2016; Mohammadi et al. 2005; Singh et al. 2005; Valipour et al. 2013). In the current research, various models will be used: rural monthly engineering model with two parameters (GR2M), Thomas ABCD model and the non-seasonal autoregressive integrated moving average (ARIMA) models, are involved to predict flow in three basins in the northwestern Algeria.

Conceptual models (occasionally called gray-box models) are transitional between theoretical and empirical models. Generally, conceptual models consider physical laws but in extremely simplified form. There is a big variety of models which belong to this class; examples which are familiar to several researchers are HBV, GR, and the Thomas ABCD model (Xu 2002). Conceptual models, especially GR2M, have been widely employed for predicting and simulating different hydrological processes: drought forecasting (Belarbi 2017), assessing hydrologic changes (Lyon et al. 2017), and quantifying the impact of human activities and climate change on stream flow (Ahn and Merwade 2014). Researchers sub-cited in references argued that this kind of models presents good capacities to reproduce the flows from the rainfall data under climate variability.

Traore et al. (2014) apply the GR2M and the GR4J models for evaluating water resources of the Koulountou River Basin, a tributary of the Gambia River where they found that the GR2M shows the best performance and easier for calibration and application, and they conclude that it can be used to restore gaps in series of flows. Bai et al. (2015) compared the performance of 12 monthly water balance models in different climatic catchments of China. They found the GR model to be more suitable for monthly river predicting. They conclude that the GR2M two-parameter model is sufficient to give a good performance for monthly runoff simulation. Sharifi (2018) used the GR2M two-parameter monthly water balance model for simulating monthly seasonal and annual runoff in Kalu and Mahaweli rivers of Serilanca. He found that the model with the simple structure and two parameters proved to be a very efficient model when simulating runoff in different time scales. It was concluded too that this two-parameter monthly water balance model can be easily and efficiently used for the water resources planning and management; this is due to its plainness and great productivity in performance.

The ABCD model is a conceptual water balance model well known in modeling different hydrological processes. Alley (1984) studied the comparison of three water balance models; the results showed that the ABCD model had the best performance, especially in its capacity to reproduce the observed behavior of ancillary catchment states (e.g., groundwater). Wang et al. (2011) used the ABCD model for simulating monthly stream flow in the Australian catchments; they found that the ABCD model is a good performer at simulating low flow compared with daily water balance model. They concluded that monthly water balance is more advantageous compared with daily water balance model for the reason that the monthly one requires less data with a low computational cost. In order to estimate the impact of climatic variation on the stream flow in the Yiluo River in China (Liu et al. 2013) carried the ABCD water balance model to see the attribution of change of stream flow, the results showed that the ABCD model proved a high accuracy for long-term simulation of stream flow. Currently, mathematical models have taken over the most important tasks of problem answering in hydrology. A mathematical model expresses the system behavior by a set of equations, perhaps together with logical declaration stating relationships between variables and parameters without taking into account the groundwater behavior of the catchment (Xu 2002). Box-Jenkins models became one of the universal models in estimating hydrological time series, particularly after publishing the work of (Series 1976). An ARIMA non-seasonal model is an overview of an ARMA model. This model is applied in some cases where data clarify suggestion of non-stationary, where an initial differencing step (corresponding to the “integrated” part of the model) can be applied one or more times to remove the non-stationary. Therefore, ARIMA models are non-static and cannot be used to estimate the missing data. However, these models are more suitable for estimating changes in a process (Karamouz and Araghinejad 2012). The non-seasonal autoregressive moving average (ARIMA) models can be employed in a varied range of requests in many hydrological modeling problems such as stream flow prediction (Al-Juboori and Guven 2016; Mohammadi et al. 2005), rainfall–runoff modeling (Castellano-Méndez et al. 2004; Farajzadeh et al. 2014; Machekposhti et al. 2018), and drought forecasting (bin Shaari et al. 2017; Mishra and Desai 2005). Ahsan and O’Connor (1994) stated that Box-Jenkins model appears quiet reliable for predicting flow time series with a minimum mean square error when the flow forecasting model is expected to be autoregressive moving average model and the corresponding flow data are free of measurement errors. Abudu et al. (2010) applied autoregressive integrated moving average (ARIMA), seasonal ARIMA (SARIMA) and Jordan-Elman artificial neural networks (ANN) models in forecasting monthly stream flow in the Kizil River, Xinjiang, China. The results indicated that ARIMA model performed similarly to ANN models. Each of these different hydrological models has its own potential, advantages, and disadvantages. Accordingly, it is important to compare them from the standpoint of these procedures in terms of prediction accuracy and efficiency on the one hand, and their implantation in different climate conditions and catchments characteristics, on the other hand. The purpose of this study is, therefore, to evaluate whether the parsimonious hydrological models are able to simulate historical stream flow time series in northern Algeria. The main objective of the research is to compare conceptual rainfall-runoff hydrological models (ABCD and GR2M) with a stochastic modeling approach (ARIMA). Relevant to this, the following research questions are put forward:

  1. (i)

    Which model will be suitable to predict flow in this special type of complex climate?

  2. (ii)

    Have geological structures of the catchments an important effect in the simulation of these models?

  3. (iii)

    Does the degree of complexity of the model affect the accuracy of the prediction?

In Algeria, GR models have been widely used (Belarbi et al. 2017; Merabet-Baghli et al. 2018; Otmane et al. 2017; Paturel et al. 2017). They gave varying results in terms of robustness according to the geographical regions of the country. On the other hand, ABCD and ARIMA models have not been tested before; hence, the importance of this work that will allow comparison of these different approaches to find out the best alternative. Flow forecasting is a very important step in addressing the information needed for hydraulic studies, engineering structures, and protection studies.

Materials and methods

Study area

Algeria is part of the semi-arid areas which cover over 40% of the world’s land surface and which are more exposed to the phenomenon of climate change (Boudjadja et al. 2003; Meddi and Hubert 2003). The management of the water resource in North Africa is more complex than it is in tropical zones due to the deficiency of perennial wadis and other readily available water sources (Souza et al. 2016). In this study, we are interested in the northwestern part of the Algerian territory. From 1970 to 2004, the country underwent the most severe drought period, with increases in the annual average temperature (Bakreti et al. 2013) and reduction of rainfall about 25% (Meddi and Hubert 2003). The location of the studied basins extends from 34°, 24′ to 35°, 48′ north latitude and 2°, 12′ to 0°, 12′ east longitude. The highest elevation of the study area is located in the Mont of Telemcen with 1614 m and the lowest is 16 m at the Mellah hydrometric station in the Tlata basin. The present research has focused on three sub-basins, two located in the Oranian coastal area (Wadi Tlata and Wadi Mellah) and the third in the Tafna basin (Wadi Chouly) (Fig. 1). Their drainage areas are 95 km2, 720 km2, and 170 km2 respectively.

Fig. 1
figure 1

Location of study basins rainfall, flow, and climate stations

Full size image

Geologically, the study area is occupied by a geological series ranging from Primary to the Quaternary period, and it is hollowed out in a material of high variable resistance. Based on the primary schisto-quartzitic substratum and secondary carbonate formations, tertiary sediments were deposited, predominantly Miocene clays, sandstones, and quaternary alluviums occupying the valley bottom and plains (Benest 1985). Heading to the Mediterranean Sea and in the east of the study area, according to Fig. 2, we notice the presence of the Large Sabkha of Oran. This Sebkha consists of a vast salty lake surrounded by successive strata to the north (Misserghin, Amria, Bou-Tlelis, etc.) and the great plain of Mleta to the south.

Fig. 2
figure 2

General map of the massif Oranian coastal

Full size image

The northern part of Algeria is characterized by a Mediterranean climate with a relatively cold and rainy winter and a hot and dry summer. The annual rainfall reaches 400 mm in the west, 700 mm in the center, and 1000 mm in the east for the coast. This type of climate is also found in the Tellian Atlas chains where we record totals ranging from 800 to 1600 mm in the eastern summits, while the values are lower in the center (700 to 1000 mm) and in the west (600 mm). In the plains of the Tellian Atlas, rainfall varies between 500 mm in the west, 450 mm in the center and 700 mm in the east. The pluviomtric regime is characterized by a decrease of about 25% since 1975 (Meddi and Hubert 2003) and confirmed by Taïbi et al. (2016). The hydrological regime is variable. The spatial evolution of the flow follows the spatial variability of rainfall. The latter decreases from the east to the west, and from the south to the north.

Figure 3 illustrates the spatio-temporal variability of rainfall in the northwestern Algeria.

Fig. 3
figure 3

Rainfall map of the study area (the National Agency of the Water resources of Algiers (Agence Nationale des ressources hydrométrique, ANRH)

Full size image

Data

The collected data concern rainfall, runoff, and temperature. Conceptual models (ABCD and GR2M) require evapotranspiration as an essential input; for this purpose, the Thornthwaite model for estimating the evapotranspiration will be used. The data used were obtained from the two agencies responsible of rainfall network, i.e. the National Agency of Water Resources (N.A.W.R) and the National Meteorological Office (NMO). The characteristic of set data are regrouped in Table 1.

Table 1 Calibration and validation periods of the models
Full size table

GR2M

The GR2M model (agricultural engineering with 2 monthly parameters) is a conceptual model with two parameters. It is developed by the CEMAGREF (France) for applications in the field of water resources and the base flow. This model has known several versions, proposed successively by (Kabouya 1990; Makhlouf 1994; Mouelhi 2003; Mouelhi et al. 2006) which made it possible to improve gradually the performances of the model. The version presented here is that of Mouelhi et al. (2006) which appears the best performing. The schematic of the GR2M is presented in Fig. 4.

Fig. 4
figure 4

Schematic of the GR2M model (Mouelhi et al. 2006)

Full size image

At the beginning, the production store has a capacity to collect the moisture defined by parameter X1. The initial level S specifies the initial state of water/moisture in the production store of X1 capacity. Rainfall (P) is distributed such that total of the rainfall increases the level in the production store (infiltration) and the rain that does not infiltrate (excess) becomes surface flow (P1). The water level in the production store also varies due to evapotranspiration (ET). The level in the storage tank is also affected by percolation (P2). The remaining water becomes the storage water level S that is used as the starting level for the next time step (month). The routed water consisted of the infiltrated water and excess rainfall. The routed water (R1) is affected by the exchange coefficient X2 that represented groundwater exchange. X1 and X2 are free parameters that are determined through calibration. According to Mouelhi et al. (2006), the routing store has a fixed capacity of 60 mm.

ABCD

The ABCD model is a conceptual model with four parameters, it presents a diverse formulation in the evapotranspiration process, and takes into account an excess of water whenever the soil moisture reservoir (S) is not full yet (Mouelhi et al. 2006). There are four parameters governing the model behavior—A controls the amount of runoff and recharge that occurs when the soils are under-saturated; B controls the saturation level of the soils; C defines the ratio of groundwater recharge to surface runoff; and D controls the rate of groundwater discharge (https://abcd.walkerenvres.com). As displayed in Fig. 5, the ABCD contains two compartments: the first one is the soil storage (S) and the second is the ground water storage (R). The upper (S) has two productions: runoff and infiltration. Therefore, the ABCD has two inputs: rainfall (P) and evapotranspiration (ET). The outputs will be the runoff (Qs), infiltration, monthly available water, real evapotranspiration (ET), satisfied soil moisture at the last of month (S), total runoff (Q), groundwater runoff (QR) and monthly groundwater storage (R) (Fig. 3). The following equations describe the catchment runoff production (storage, soil moisture and ground water).

Fig. 5
figure 5

Schematic of the ABCD model (Thomas 1981)

Full size image
$$ {\displaystyle \begin{array}{c}{\mathrm{Storage}}_t+\sum {\mathrm{Outflow}}_t={\mathrm{Storage}}_{t-1}+\sum {\mathrm{Outflow}}_t\\ {}{\mathrm{SoilMoisture}}_t+{ET}_t+{\mathrm{Runoff}}_t+{\mathrm{Recharge}}_t={\mathrm{SoilMoisture}}_{t-1}\\ {}{\mathrm{Groundwater}}_t+{\mathrm{Discharge}}_t={\mathrm{Groundwater}}_t+{\mathrm{Recharge}}_t\end{array}}+{\mathrm{Precip}}_t $$

Box-Jenkins model

According to the previous works of Yule and Wold (Wold 1938; Yule 1926), Box-Jenkins (Series 1976) contracted a convenient approach to constructing ARIMA models, which have the basic effect on the forecasting applications and the time series analysis. As demonstrated by Valipour et al. (2013), such a model can be created using a combination of moving average (MA) and AR processes. To apply the ARIMA model, some processes have to be realized first. It proposes a three-phase practical technique to achieve a suitable model; this technique is based on identification, estimation, and diagnosis (Fig. 6).

Fig. 6
figure 6

Schematic of the ARIMA forecasting model (Series 1976)

Full size image

The first step is to check the stationarity of the data set. “Stationarity” designates that the series remain at a constant level over time. Data should also show a constant variation in their variability over time. This is easily appreciated with a series that is spectacular seasonal and grows at a faster rate. Many accounts associated with the process cannot be determined without gathering the stationarity conditions. The ARIMA model function is represented by (p, d, q); these parameters are successively the order of the autoregressive components, the number of differencing operators, and the highest order of the moving average term. According to Series (1976), the Box-Jenkins methodology proposed an extensive variety of ARIMA models to select from; for the present study, the projected ACF and PACF were used to select one or more applicable ARIMA models. The basic idea is that each ARIMA model should be linked with theoretical ACF and PACF. During the identification step, we obtained the estimated ACF and PACF, deliberate from the data derived from different theoretical ACF and PACF. Consequently, the most resembled model to the theoretical one will be chosen, this model is then adapted to the available data by calculating the essential coefficients. If these estimated coefficients do not answer some mathematical conditions, the model is rejected (Valipour et al. 2013).

In time series predicting, Yt, can be reflected as the output of the linear system, where the input εt is a Gaussian white noise, Yt is defined by Eq. (1) (Hamilton 1994);

$$ {Y}_t=\upmu +\psi \left(\mathrm{B}\right){\varepsilon}_t $$
(1)

μ is the mean of Yt, the back shift operator B is defined by Eq. 2; Bkεt = εt − k, for K = 1, 2 …, and ψ(B) is the transfer function, which links the input, εt, to the linear system output, Yt, such as ψ(B) = 1 + ψ1B + ψ2B2….

The autoregressive (AR) one is a particularly interesting class of these models. In such a model, every value Yt is the weighted and finite sum of previous values plus a random term εt. The model is then designated by AR (p), p is the corresponding order. Therefore, for μ = 0, it is expressed as follows (Valipour et al. 2013):

$$ \phi (B){Y}_t={\varepsilon}_t $$
(2)

Where ϕ(B) = 1 − ϕ1B − ϕ2B2 − … − ϕpBp is the AR (p) operator, which is a p order polynomial in B, which converges for |ϕi| < 1, the moving average (MA) approach is another interesting model to insure the stationary conditions, in which each value is the sum of q + 1 previous values of a white noise, hence Yt is defined in the following equation (Eq. 3) (Valipour et al. 2013);

$$ {Y}_t=\theta (B){\varepsilon}_t $$
(3)

Where θ(B) = 1 − θ1B − θ2B2 − … − θpBp is the MA (q) operator, which is a q order polynomial in B, and it converges for |θi| < 1 in order to insure for invertibility condition insurance (Valipour et al. 2013).

The linear combination of AR (p) and MA (q) models generate a new mixed process called autoregressive and moving average of order p and q, ARMA (p, q) and it is expressed as follows (Valipour et al. 2013).

$$ \phi (B){Y}_t=\theta (B){\varepsilon}_t $$
(4)

In this case, Yt, can be considered as the output of a linear filter whose transfer function is the ration of the two polynomials θ(B) and ϕ(B):

$$ {Y}_t=\frac{\theta (B)}{\phi {(B)}_t}{\varepsilon}_t $$
(5)

This type of modeling assumes data to be stationary (Mohammadi et al. 2005), but this is not the case in most of hydrological time series (Hamilton 1994). In this case, it is always possible to make them stationary using some mathematical transformations such as the differences and the normal logarithm. Hence, ARMA model becomes ARIMA; more details on the method can be found in several references such as (bin Shaari et al. 2017; Mohammadi et al. 2005; Valipour 2015; Valipour et al. 2013). The general basis of the used methods in this investigation revolves around the identification of the hydrological models that best fit each catchment considered (Hamilton 1994). The assessment of the goodness of fit for the three models was mainly performed using the ratings suggested (McCuen 2016); Pearson’s correlation coefficient between observed and predicted runoff in the studied period was also analyzed. The conceptual hydrological models are defined with two parameters for the GR2M and four parameters for the ABCD, they present different moisture balances according to the different processes in a hydrological system through all phases of the hydrological cycle. The ARIMA model with three parameters takes into account only the observed runoff as the exclusive variable as a time series; the goodness of fit of this type of model based on the Akaike’s Information Criterion (AIC) and the Bayesian Information Criterion (BIC), depends also on the coefficient of determination R2.

Goodness of fit

The model parameters were calibrated using different efficiency indices such as the Nash-Sutcliffe Efficiency (NSE) (Nash and Sutcliffe 1970), the coefficient of determination R2, the root mean square error (RMSE) (Singh et al. 2005) and the percent bias (%). Moriasi et al. (2007) proposed a rating system to classify the models into four types: very good, good, satisfactory, and unsatisfactory.

For the Box-Jenkins model, to analyze the performance of these models, another two popular measures were applied such as the Akaike information criterion (AIC) and the Bayesian information criterion (BIC). They are used for comparing maximum likelihood models.

$$ NSE=1-\frac{\sum \limits_{i=1}^n{\left({Q}_{oss,i}-{Q}_{\operatorname{mod},i}\right)}^2}{\sum \limits_{i=1}^n{\left({Q}_{oss,i}-{\overline{Q}}_{oss}\right)}^2} $$
(6)
$$ RMSE={\left[\frac{1}{n}\sum \limits_{i=1}^n{\left(Q\operatorname{mod}{,}_i- Qoss,i\right)}^2\right]}^{1/2} $$
(7)
$$ BIAS\left(\%\right)=\frac{\sum \limits_{i=1}^n\left({Q}_{\operatorname{mod},i}-{Q}_{oss,i}\right)}{\sum \limits_{i=1}^n{Q}_{oss,i}}\ast 100 $$
(8)
$$ AIC=-2\ast \mathit{\ln}\left(\mathrm{likelihood}\right)+2\ast k $$
(9)
$$ BIC=-2\ast \mathit{\ln}\left(\mathrm{likelihood}\right)+\mathit{\ln}(N)\ast k $$
(10)

Where n is the number of data observations, Q (oss) is the observed stream flow, Q (mod) is the estimated stream flow, and \( \overline{Q} \)(oss) is the average of observed stream flow. The modeling procedure having lower RMSE values can be supposed to be the most accurate model for stream flow prediction. A value of NSE equals to 1 means an optimal performance of the model. When the simulated values correspond exactly to the observed values, I, t established the model’s ability to simulate the flow values from the mean values. Correlation coefficient values near unity indicate a linear relationship between the observed and predicted stream flows (Table 2).

Table 2 Range of coefficient of efficiency NSE, BIAS (%), and coefficient of determination (Moriasi et al. 2007)
Full size table

Results and discussion

Calibration and validation of the models

The GR2M model

The calibration of the GR2M model is based on two parameters: X1 and X2, coefficient of production and exchange, respectively. The calibration of the model was done by comparing the observed and predicted stream flow data in the three catchments for different periods (Table 1). All the calibrated parameters of the selected model were identified using iterative procedure for several times based on high value of R2 and NSE criteria and low value for RMSE. Table 3 presents the calibrated parameter values for the model considering a high value for R2 and for the NSE criteria. Furthermore, a graphical verification was done to ensure the reliability of the rainfall–runoff behavior in catchments. The performance of the GR2M model in the validation period was evaluated using a graphical illustration to compare the observed and simulated flows (Fig. 7).

Table 3 Calibration and validation performance for GR2M model
Full size table
Fig. 7
figure 7

The comparison of the observed and predicted monthly stream flow using the GR2M procedures for Chouly and Tlata wadi watershed for validation

Full size image

Considering Fig. 7, it can be assumed that there is a relatively good agreement between the observed and simulated flows. A comparison of the observed and simulated monthly flows for the GR2M model is shown in Fig. 8, which confirms the robustness of the calibrated model to predict flow (Table 3).

The ABCD model

For predicting runoff using the ABCD model, a process was developed for calibrating the model automatically and for predicting runoff. For each catchment, the spatial average of monthly rainfall and PE is computed over the entire studied period, the model was used with these data as input, while their respective four parameters were improved within a defined range (Table 4). The objective of these systematic test runs was to optimize the statistical criterion between the predicted and observed values of flows for the calibration and the validation period. In order to obtain a successful calibration situation by using automatic optimization procedures, it will be necessary to formulate numerical performance measures that reflect the prediction objectives. Calibrated and validated values for the model were presented in Table 4, where results showed that in calibration period the agreement between the observed and predicted monthly flows is widely satisfactory.

Table 4 Calibration and validation performance for ABCD model
Full size table

The calibration performance was evaluated using plots of the observed and predicted data series and analysis of the statistical measures. Table 4 presents the calibration and the validation performance of the ABCD model based on the most accurate model run taking into account the best fit of the model’s parameters. As can be seen in Table 4, the agreement between the observed and simulated monthly flows is generally more satisfactory in the calibration period compared with the validation period. The performance of the ABCD model in the validation period was evaluated using a plot of the observed and predicted flows (Fig. 9). It can be noted that the ABCD model predicted the flow better in the Tlata basin than Chouly basin. As a result, the underestimation of the flow by the ABCD model in Chouly wadi could be seen in Fig. 10. In addition, Fig. 10 shows a comparison of the observed and simulated monthly stream flows with respect to the 1:1 line.

Fig. 8
figure 8

The comparison of the observed and predicted monthly flow using the ABCD model for Chouly and Tlata basins in validation

Full size image
Fig. 9
figure 9

The scatter plots of the observed and predicted monthly flows using the GR2M in the Chouly and Tlata wadi watersheds for the validation data set

Full size image
Fig. 10
figure 10

The scatter plots of the observed and predicted monthly flows using the ABCD model in the Chouly and Tlata basins for the validation period

Full size image
Fig. 11
figure 11

The RACF for the selected ARIMA (1, 1, 1) model; (a) display of the quality of the calibration (b) display of the quality of the validation

Full size image
Fig. 12
figure 12

Results of calibration obtained by the ARIMA (1, 1, 1) model

Full size image
Fig. 13
figure 13

The scatter plots of the observed versus the predicted monthly stream flows using the ARIMA model in the Mellah wadi watersheds in the validation data set

Full size image

ARIMA models

There are five stochastic models to model the monthly flows with the Box-Jenkins model. The most suitable one will be selected based on the Akaike’s information, the Bayesian criterion and the root mean square error. The lesser is the RMSE, AIC, and BIC, the better is the model. The monthly time series need some transformations to be stationary as indicated in “Box-Jenkins model.” Figure 11 shows the autocorrelation function and the partial correlation function that are applied on data. The first two displays show the ACF and the PACF for initial non-stationary data before application of the first-order transformation (d = 1). Figure 11, of ACF and PACF after transformation of data, confirms that the data are compliant to be used in ARIMA model where we have a very significant value in lag one for the ACF and two significant values in the first lags in PACF.

In this way, data are ready for an ARIMA modeling with (d = 1), several researchers such as Hamilton (1994), Mohammadi et al. (2005), and Valipour et al. (2013) say that an order of differences of more than one is not recommended for time series because data with (d > 1) results have to be further from reality and the model to be overestimated even with this order of difference we got the perfect correlation between the observed and the predicted observations.

As shown in “Box-Jenkins model,” three steps of modeling were followed for selecting the most appropriate model among the different ARIMA models to the time series of stream flows in the three catchments located in the study area. To define the insistence system in the series the ACF and the PACF were tested. Different models were identified based on the ACF and PACF analysis. Table 5 presents the 9 selected ARIMA models which might be suitable for predicting the monthly stream flow in the three catchments under study. For selecting the best-fit model out of the various tested models, the AIC and BIC criteria were used; however, the BIC was intended to select one of the competitive models. The ARIMA (2, 1, 3), ARIMA (1, 1, 1) models performed better than other models. The ARIMA (1, 1, 1) model had the lowest value of BIC in the three cases of the study (Table 5).

Table 5 Calibrated model parameters for the Box−Jenkins ARIMA models
Full size table

The residual autocorrelation function tests were employed for checking the adequacy of the models fitted to the time series. Depending on the RAFC tests and residual correlation, the ARIMA (1, 1, 1) model was the most suitable one. Figure 11 shows a plot of the RACF for the selected ARIMA (1, 1, 1) model in the three catchments under study. It can be noted that the estimated values fall within the 95% confidence interval. Therefore, the residuals are normally distributed and white noise. The final calibrated model must be tested for the next step to predict stream flow data series in the period of validation. The selected ARIMA (1, 1, 1) model predicted moderately the stream flow in the three catchments in the validation period. Figure 12 shows the comparison between the observed and predicted stream flows in the validation step. However, the underestimation of the stream flow by the selected model during high flow periods is seen in Fig. 12 especially for the Tlata wadi. Figure 13 shows the accuracy of the ARIMA model in the validation stage as a scatter plot of the observed and predicted stream flows. Figure 12 shows the calibration and the validation with the Box-Jenkins ARIMA model with one parameter of autoregressive and moving average. It was the simplest and the quickest to predict stream flows in a semi-arid catchment. It is so clear also that the Box-Jenkins model is more powerful to predict low flow, but it was so weak in high flow or peak point such as extreme events that are often observed in the area of study. On the other hand, the trend period has no influence during the simulation and the model is more stable after some iterations.

Hydrological models comparison

Table 6 presents the comparison between the stochastic model (ARIMA) and the conceptual models (GR2M and ABCD) in the validation stage based on different statistical indexes namely, RMSE, NSE, BIAS, and R2.

Table 6 The comparison of different hydrological models based on accuracy measures
Full size table

According to these criteria, the GR2M model performed better than the other models. The ABCD model gave competitive results of prediction compared with the ARIMA model in the case of Chouly and Tlata wadi but it is not the case in the third catchment where the ARIMA model prove its ability for modeling stream flow in the Mellah wadi. GR2M and ABCD models show small values of prediction errors with reference to the RMSE criterion. It can be noted from Table 5 that the statistical evaluation of the predicted stream flow provided the less values of RMSE for the ABCD model in the three catchments under study. On the other hand, the GR2M model gave little more value than the ABCD model, 0.34, 0.46 respectively for the Chouly and Tlata Wadi, noting that the Mellah wadi is a special case regarding the complexity of the physical behavior of the catchment. This indicates that the stream flow prediction of the ABCD gave less error than those of the GR2M model. Based on the RMSE criteria the GR2M and the ABCD performed better than ARIMA model in the Chouly and Tlata wadi.

The BIAs index was used to evaluate the quality of prediction for the different models used in the paper. The ABCD model shows least values of BIAS of 7.51 and − 8.48 for the Tlata and Chouly wadi respectively.

This result shows the small range of overestimation and underestimation in stream flow prediction using the ABCD model. Then, again, the results given by the GR2M model show the large underestimation in stream flow prediction in the Tlata wadi and the high overestimation in both of Chouly and Tlata wadi (Table 5), this value is still in an acceptable standard (Moriasi et al. 2007). However, the accuracy measures for the ARIMA model in Tlata wadi showed a relatively acceptable performance (RMSE = 0.407, NSE = 0.68, BIAS =− 13.29, R2 = 0.73). Nevertheless, it was not the case for the two other models in this catchment. According to the scatter plots of the predicted and observed stream flow for each model in previous sections, it can be noted that the GR2M model has the superiority of predicting in respect to the 1:1 line. Model parameters for the GR2M appear similar to all the analyzed catchments. The ABCD parameters are instead rather variable for the three catchments. This circumstance is probably caused by the adaptation of the different hydrological processes embedded in the model structure to the different cases of the study. What appears quite clear is the inability of the ABCD model to capture and describe the groundwater processes (d = 0), which is of particular importance for the cases studied.

For evaluating the prediction accuracy for the models, the NSE criterion was used; the NSEs of the ABCD and the ARIMA model were ranging between 0.54 and 0.66 in the Chouly and Tlata wadi catchment. The results reveal that the simulation of these models is not satisfactory but still in an acceptable standard. On the other hand, the GR2M model showed a satisfactory prediction accuracy founded on the NSE criteria with an average value superior than 0.70 which sets the GR2M model better than the other two models. This implies that the GR2M model is the most accurate modeling procedure based on NSE criteria.

For the complex catchment, which is the Mellah wadi, the ARIMA proves good ability for modeling this type of catchments with NSE = 0.68, where the input data were confined only on stream flow as predictor, as indicated in “Study area.” This basin is characterized by a specific geology represented, in certain areas of the basin, by Karsts. Many authors have found that the ARIMA model gives good results in basins with a similar geology (Ghanbarpour et al. 2010; Graupe et al. 1976; Pallard et al. 2009).

The correlation coefficient R2 was used to compare the different models based on the quality of the agreement between the observed and predicted stream flow. The average value of the correlation coefficients for the GR2M and ABCD models were 0.70 and 0.60, respectively (Table 5), which shows a good agreement between the observed and simulated stream flow values. On the other hand, the ARIMA model shows a satisfactory agreement and correlation in the Chouly and Tlata wadi catchments and a good agreement in the Mellah wadi catchment. This result confirms the robustness of the prediction from the GR2M and the ABCD modeling procedures for stream flow simulation in the study area. However, the GR2M model accurately predicted the stream flow in high flow and low flow. The ABCD model performed very well but not as accurately as the GR2M model. Yet, the ARIMA model predicted the flow rates very well in the Mellah wadi catchment.

Conclusion

In order to identify a valid modeling tool for stream flow predictions in the northernwest of Algeria, conceptual (GR2M and ABCD) and stochastic (ARIMA) modeling approaches have been investigated and compared in this research. The different approaches have been used to simulate stream flow at three catchments featured by different properties, particularly in terms of the hydrogeological aspect. In fact, among the three, the Mellah catchment represents the most complex catchment because of important and diffused Karst phenomena.

As an overall finding, all the applied methods are characterized by similar hydrological performance in terms of goodness of fit indices, although a rank in model performance can be identified. Conceptual models appear indeed to moderately outperform stochastic models. In particular, the GR2M has been identified as the best modeling solution for the case study, probably due to the fact that this specific approach is able to model the water exchange term with neighboring catchments. This is done through the parameter X2 which, according to the values found during the calibration period, leads to effectively important contributions of groundwater exchange. The second best model for the case study is represented by the ABCD model which, differently to the GR2M model, was not able to properly characterize the groundwater store. This can be noted in the results of the parameters calibration which has led to a value of zero for the parameter d that accounts for this particular hydrological process. The poor performance of conceptual models in the case of the Mellah catchment can probably be ascribed to its relevant hydrogeological processes which cannot be correctly identified and described by the conceptual parameters.

As a conclusion, this study can be considered as the first step toward a regional scale analysis where the ability of each of the considered models may be validated over a larger number of catchments and where furthermore an investigation of regional model parameters can be achieved.