1 Introduction

Extensive hydrological data is necessary for the planning and management of water resources. However, one of the major challenges facing hydrologists is a lack of hydrological data at a suitable spatiotemporal scale. This matter was highlighted by the International Association of Hydrological Sciences (IAHS) declaration of the decade (2003–2012) for Predictions in Ungauged Basins (PUB). The declaration promoted the formulation and implementation of appropriate programs to engage the scientific community to improve the capacity to make accurate predictions in ungauged basins (Sivapalan et al. 2003). Investigating the relationship between hydrological responses and a spatial feature of a given catchment can contribute to an understanding of the knowledge needed to improve the capacity of a given model in order to predict hydrological responses in ungauged catchments.

Conceptual rainfall-runoff models in general and Tank model in particular, which their applicability has been extending from prediction of stream flows (Ngoc et al. 2016) into that of groundwater (Aqili et al. 2016 and Hong et al. 2016) and flood forecasting tasks (Sugiura et al. 2016), are often useful when compared to physically based models due to their structural simplicity and low computational requirements. Kim (2014) conducted a study to investigate the predictive capacity of a rainfall-runoff model for multiple objectives on the adequacy of regional relationships. In this study an effort was made to regionalize the hydrological response of catchments using three different features of catchments, namely: area, slope and effective depth of soil. Achieving accurate predictions can be difficult when applying a hydrological model with a good fit for one catchment to another, since it must be recalibrated and validated against flow data from the new catchment (Gargouri-Ellouze and Bargaoui 2009; He et al. 2011). The concept of regionalization, by which some common information is transferred from one catchment model to another within a certain homogenous geographic area, has been introduced to overcome this issue and to achieve more accurate flow predictions for ungauged catchments. There are different methods for conducting regionalization among which the regression analysis-based method is the most commonly applied for transferring information from gauged to ungauged catchments (Das et al. 2008; Kokkonen et al. 2003; Pandey and Nguyen 1999; Post and Jakeman 1996; Sefton and Howarth 1998; Tung et al. 1997; Yokoo et al. 2001).

Although the concept of applying catchment features in the regionalization of conceptual rainfall-runoff models began over three decades ago, the use of landscape metrics in explaining variations in different hydrological processes relying on change in spatial attributes (e.g., Uuemaa et al. 2005; Moreno et al. 2007; Amiri and Nanake 2009) of a given catchment only began in 2010. Hence, this approach is still in the early stages of its development.

Structure, composition and configuration are key attributes by which a given landscape is analyzed and described (Forman and Godron 1986). Evidence (Amiri and Nanake 2009; Turner and Rabalais 2003; Uuemmaa et al. 2007; Wickham et al. 2003; Griffith 2002) has suggested that these key attributes of a given landscape, by which shape and spatial arrangement of landscape elements (including patch, corridor and matrix) are described, have significant influence on water and nutrient flow in catchments (King et al. 2005; Uuemmaa et al. 2007; Uuemaa et al. 2005; Xiao and Ji 2007). To our knowledge, it should be noted that no studies have been conducted that link landscape metrics with hydrological responses of catchments for water quantity. The objectives of this study were thus to 1) to determine if any significant relationship exists between hydrological process-related parameters of the Tank model and landscape metrics including patch size and patch shape (mean shape index, mean perimeter-area ratio) of land use/land cover and soil types; 2) to specify to what extent a regression analysis-based regionalized model is reliable by using uncertainty analysis.

2 Materials and Methods

2.1 Study Site

The study was carried out in Germany (Fig. 1) where thirty catchments with available meteorological, water discharge, soil and land use/ land cover data were selected. The area of the catchments varies between 53 and 737 km2. Geographical locations, hydro-climatic features of the catchments are summarized in Table 1 and Fig. 2.

Fig. 1
figure 1

Geographical location of river discharge gauging stations in the study area

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Table 1 Geographical situation, hydro-climatic, physical features of the catchments
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Fig. 2
figure 2

Proportion of land use/land cover (%) in the catchments of the study area

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2.2 Data Sets

Daily mean discharge data for the catchments were obtained from the German Federal Institute of Hydrology (2009) and daily mean precipitation, temperature and sunshine duration for the years 1990–2004 were obtained from the German Meteorological Service (2010). For each gauging station, boundaries of the upstream catchment and river networks were downloaded on April 10, 2009 from (European Commission Institute for Environmental Sustainability 2009) (Fig. 1). Geometric features of each catchment including area, area-perimeter ratio, gradient, compactness coefficient, and drainage density were then calculated using the Spatial Analyst tool in ArcGIS 9.2.

CORINE Land Use/ Land Cover (LULC) maps (1990 and 2000) were overlaid on catchment boundaries to determine the proportion (%) of different LULC types in each catchment and to calculate the landscape metrics (European Environment Agency 1999). It should be noted that for each individual catchment, referring to the years of availability of its hydrological data either CORINE LULC 1990 or CORINE LULC 2000 was chosen as its land use/land cover data. The following eight LULC classes were used to describe land use/land cover of the catchments: urban, agricultural, grassland, broad-leaved forest, needle-leaved forest, mixed forest, water bodies, and ‘other’. Landscape metrics of these LULC classes were then calculated by applying the Patch Analyst 3.1 extension (Elkie et al. 1999) in ArcView GIS 3.2 (Mitchell 1999; ESRI 1999) (Fig. 2).

Catchment soil layer information was extracted from a study conducted by Ballabio et al. (2016). This data was originally based on LUCAS 2009 Topsoil 20,000 sample point data with a 500 m resolution.

2.3 Methods

A classical two-step approach was employed for the regionalization of the conceptual rainfall-runoff model. The model was first calibrated for a number of catchments, and the optimized parameter model was then linked to catchment attributes using regression analysis (Hundecha and Bárdossy 2004). The Tank model (Sugawara et al. 1984) was individually applied to 30 catchments in order to determine the optimized values of its parameters. The Nash-Sutcliffe efficiency index (EI), which varies between − ∞ and 1, was applied to evaluate the performance of the optimized model for each catchment using:

$$ EI=\frac{{\displaystyle {\sum}_{t=1}^T{\left({Q}_0^t-{Q}_m^t\right)}^2}}{{{\displaystyle {\sum}_{t=1}^T\left({Q}_o^t-{\overline{Q}}_o\right)}}^2} $$
(1)

where Q o is the observed parameter, in this case discharge (m3 s−1), Q m is the predicted discharge (m3 s−1), and \( {\overline{Q}}_o \) is the mean observed discharge (m3 s−1).

2.3.1 Tank Model

The Tank model, proposed by Sugawara et al. (1984), is a simple rainfall-runoff model that is composed of a number of tanks laid vertically in series (Ishihara and Kabatake 1979). A standard Tank model consists of four tanks and five outflows (Fig. 3). Rainwater enters through the top tank and then the water is partially discharged from each subsequent tank through side-outlets or by infiltrating and percolating downward to the bottom outlet of a given tank, and on to the next lowest tank. River discharge can be calculated by summing the outputs from the side-outlets (Sugawara et al. 1984).

Fig. 3
figure 3

General scheme of the Tank model indicating all parameters of the model and flow components

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The tank model is based on the following catchment-scale water balance equation:

$$ \frac{dH}{dt}={P}_{(t)}-E{T}_{(t)}-{Y}_{(t)} $$
(2)

where H is the height of water (mm), P is the depth of rainfall (mm), ET is the amount of evapotranspiration (mm), Y is total flow (mm d−1), and t is time, in days.

In the Tank model, total flow is the sum of the horizontal flow of all tanks:

$$ Y(t)={Y}_a(t)+{Y}_b(t)+{Y}_c(t)+{Y}_d(t) $$
(3)

where Y a (t), Y b (t), Y c (t), Y d (t) are the horizontal flows of each of the four tanks, progressing vertically downward (Fig. 3).

The water balance in each tank can be written as:

$$ \raisebox{1ex}{$d{H}_a$}\!\left/ \!\raisebox{-1ex}{$dt$}\right.=P(t)-Et(t)-{Y}_a(t) $$
(4)
$$ \raisebox{1ex}{$d{H}_b$}\!\left/ \!\raisebox{-1ex}{$dt$}\right.=Y{a}_0(t)-Yb(t) $$
(5)
$$ \raisebox{1ex}{$d{H}_c$}\!\left/ \!\raisebox{-1ex}{$dt$}\right.=Y{b}_0(t)-Yc(t) $$
(6)
$$ \raisebox{1ex}{$d{H}_d$}\!\left/ \!\raisebox{-1ex}{$dt$}\right.=Y{c}_0(t)-Yd(t) $$
(7)

where Ya, Yb, Yc and Yd represent the horizontal flow components of tanks A, B, C and D, respectively, and Ya 0, Yc 0 and Yb 0 represent the vertical flow components (infiltration and percolation) of each tank (A, B and C).

The Tank model parameters can be classified into two types: i) those which reflect hydrological processes (a 0 , a 1 , a 2 , b 0 , b 1 , c 0 , c 1 and d 1 ), and ii) those which reflect water levels (Ha 1 , Ha 2 , Hb 1, and Hc 1 ) in the four tanks and serve as initializing parameters in the model. The hydrological process-reflecting parameters can further be divided into: i) those representing vertical movements of water into and through the soil matrix (e.g., infiltration and percolation—a 0 , a 0 and c 0 ), and ii) those representing horizontal movements of water at the soil surface as runoff (a 2 ), in sub-surface soil as sub-surface flow (a 1 ), in deep soil layers as intermediate flow (b 1 ), and in geological formations as sub-base flow and base flows (c 1 and d 1 ).

Calibrating the Tank model for a given catchment requires input data including daily mean stream discharge (mm d−1), daily rainfall (mm d−1) and real evapotranspiration (E v; mm d−1). For each of the catchments, potential evapotranspiration was initially estimated using the Hargreaves Model:

$$ E{T}_0=0.0135\left(T+17.78\right){R}_s $$
(8)

where ET 0 is the potential evapotranspiration (mm d−1), Rs is the solar radiation equivalent evaporation (mm d−1), from which sunshine duration was estimated (FAO 1998), and T is mean temperature (°C).

For estimation of the real evapotranspiration, ET 0 was then adjusted using:

$$ {E}_v={K}_c\times E{T}_0 $$
(9)

where, E v stands for real evapotranspiration (mm d−1), Kc is the crop or LULC coefficient, whose values for the CORINE LULC classes were based on values cited in the literature (Allen et al. 1998; Heryansyah 2001; Verstraeten et al. 2005). For this study, the tank model optimizer application (Yanto and Setiawan 2003; Setiawan et al. 2003) was applied to optimize the parameters of the Tank model for each of the catchments.

2.3.2 Landscape Metrics and Soil of the Catchments

Soil type of the catchments were classified based on USDA soil texture classification and proportion (%) of the soil types was calculated (Table 2).

Table 2 Proportion (%) of soil type of the catchments
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Landscape metrics including mean patch size (mps),Footnote 1 mean shape index (msi),Footnote 2 mean perimeter-area ratio (mpar)Footnote 3 and patch density (pd)Footnote 4 as a percentage were calculated for the LULC of each catchment at the class-level by Patch Analyst 3.1 (Elkie et al. 1999) (Table 3).

Table 3 Descriptive statistics of landscape metrics of the catchments
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2.4 Statistical Analysis

The optimized parameters of the Tank model were considered as dependent variables. Table 4 indicates the statistics of the optimized parameters of the Tank model.

Table 4 Descriptive statistics of the optimized parameters of the Tank model for the catchments
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Stepwise multiple regression was carried out to develop the best fit model for specific optimized Tank model parameters (dependent variable), vs. landscape metrics of the LULC types and soil types (independent variables):

$$ {y}_i={\beta}_0+{\beta}_1{x}_1+{\beta}_2{x}_2+\dots +{\beta}_{n-1}{x}_{n-1}+{\varepsilon}_i $$
(10)

where y i is the optimized parameter of the Tank model, x 1.... x n−1 are the landscape metrics of the LULC types and soil types of the catchments, β 0 …β n−1 are coefficients, and ε i is constant of the model.

For each regression model, the initial independent variables were the LULC-related landscape metrics including: mean perimeter to area ratio ( mpar ), mean patch size ( mps ), mean shape index ( msi ), patch density ( pd ) for a given LULC class namely: urban area (U), agricultural land (A), broad-leaved forest (F), needle-leaved forest (N), mixed forest (M), grassland (G) and water bodies including moors and peat lands, inland marches, peat bogs (W), and compositional metrics (%) of soil types. The optimized parameters of the Tank model were also considered as preliminary fixed dependent variables (Fig. 3) which include: infiltration coefficient (a 0 ), sub-surface flow (a 1 ), surface flow (a 2 ), intermediate flow (b 1 ), percolation coefficients (b 0 ) and (c 0 ), sub-base flow (c 1 ), base flow (d 1 ), and the water storage level of surface, sub-surface, intermediate and sub-base flows (Ha 1, Ha 2 , Hb 1 and Hc 1 , respectively).

Inter-variable collinearity of the regression models was evaluated using the Variance Inflation Factor (VIF), where a value of VIF < 10 means that the variable was set aside from further steps in the modeling process (Neter et al. 1996).

For a given optimized parameter of the Tank model, the appropriate model was selected based on the regression statistics of r 2 and p-value (i.e., significance of the regression model). Finally, the goodness-of-fit of statistically significant regression models was evaluated graphically using scatter plots. Statistical analyses were completed using Excel Add-ins (XLSTATTM 2006) and SPSS for Windows Release 11.5. For all the catchments, the Tank model was executed to achieve the optimized parameters of the model on a daily basis by the procedure imbedded in the model program. The model was calibrated for the period of 1990–2001, and validated for the period of 2002–2005 (Table 5).

Table 5 Nash -Sutcliffe coefficients for calibration and validation periods for the catchments
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2.5 Uncertainty Analysis

Determining the uncertainty of a model is a very important step in any given modelling process (Wagener and Montanari 2011). It helps to better understand how a model will behave under different conditions. Therefore, to clarify how the developed models in the present study might act when applied to predict the optimized parameters of the Tank model, an uncertainty analysis was conducted by applying the Monte Carlo simulation approach (Amiri et al. 2012). The approach includes the following steps: i) defining the model; ii) determining the statistics of central tendency and theoretical distribution of dependent and independent variables; iii) generating random data (15,000 in the case of our study); iv) sampling from the generated data; v) simulating the model responses using the generated data; and vi) inferring probabilistic behaviours of the model of interest. The latter is conducted by determining the cumulative distribution function, which is drawn for the simulated outputs of the model of interest.

3 Results and Discussion

Area, slope and drainage density of the catchments were found to vary between 53 and 573 (km2), respectively (Table 1). On average, 43.6 % of catchment areas were covered by different types of forests (Fig. 2), namely broad-leaved forest (9.4 %), needle-leaved forest (24.1 %) and mixed forests (10.1 %). Farmlands (27.6 %) were ranked second in terms of LULC components. Grasslands and semi-urbanized areas represented 14.4 and 4.9 % of catchment areas, respectively. The fraction of the catchment areas that are covered by water bodies was 1.1 (%). The regression models (r 2 ≥ 0.35), which have been developed to link the optimized parameters of the Tank model with the catchment landscape attributes, are as follows:

$$ ln{a}_0 = - 1.683+0.285 \ln SiL+0.362 \ln {U}_{pd}-0.105 ln{W}_{pd} $$
(11)
$$ ln{a}_1=-2.589-13.899{N}_{pd} $$
(12)
$$ ln{b}_1=-0.396-0.077\ LS+0.000{N}_{mpar}-3.696{F}_{pd}-0.013\ SiL $$
(13)
$$ {d}_1=0.000+0.000 ln{F}_{pd}+0.000 \ln SiL $$
(14)
$$ lnH{a}_2=5.152-0.244{M}_{msi}-0.010\ L+0.007{W}_{mps} $$
(15)
$$ \begin{array}{l}H{b}_1=3.335-2.015 \ln LS+3.656\ ln\ {W}_{msi}-0.234 \ln {W}_{mps}-0.267 \ln {W}_{mpar}+\hfill \\ {}0.775 \ln {G}_{pd}-0.570 \ln {A}_{pd}\hfill \end{array} $$
(16)

where,

a0 :

is the infiltration coefficient,

a1 :

is the subsurface flow,

b1 :

is the intermediate flow,

Ha2 :

is the water storage level of sub-surface flow,

Hb1 :

is the water storage level of intermediate flow,

SiL:

is the percentage of the catchment areas, which is covered by silty loam soil,

LS:

is the percentage of the catchment areas, which is covered by loam silt soil,

L:

is the percentage of the catchment areas, which is covered by loam soil,

Wmps :

is the mean patch size of the water body patches,

Mmsi :

is the mean shape index of mix forest patches,

Nmpar :

is the mean perimeter to area ratio of needle-leaved forest patches,

Wmapr :

is the mean perimeter to area ratio of water body patches and

Upd, Wpd, Npd, Fpd and Gpd :

are the patch density of urban, water body, needle-leaved forest, broad-leaved forest and grassland patches, respectively.

According to Fig. 3, the findings of the present study can be classified in accordance with the types of the parameters of the Tank model as follows.

3.1 Vertical Movement of Water- Reflecting Parameters

Vertical movement of water-reflecting parameters include the infiltration coefficient (a 0 ) and the percolation coefficients (b 0 ) and (c 0 ). Table 6 indicates that mean patch size of water bodies (W mps ), and patch density of urban area (U pd ) and water bodies (W pd ) were selected for use in regionalized regression models. An inverse relationship was observed between the patch density of water bodies and the mean patch size of needle-leaved forest patches with the values of infiltration coefficient (a 0 ), and the percolation coefficients (c 0 ) in the catchments, respectively. This relationship implies that increasing the patch density of water bodies and the mean patch size of needle-leaved forest patches can result in decreasing infiltration coefficient (a 0 ) and percolation coefficients (c 0 ) in the catchments. However, the patch density of urban areas and the mean patch size of water bodies are positively related to infiltration coefficient (a 0 ) and percolation coefficients (b 0 ). Accordingly, if the patch density of an urban area and mean patch size of water bodies increase, then infiltration coefficient (a 0 ), and percolation coefficients (b 0 ) will consequently decrease. It is important to note that none of the compositional metrics (%) of soil types except silty loam (SiL) are introduced in the regionalized models.

Table 6 Result of regression analysis-based regionalization between the optimized parameters of the Tank model, landscape metrics and soil types
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3.2 Horizontal Movement of Water- Reflecting Parameters

Sub-surface flow (a 1 ), surface flow (a 2 ), intermediate flow (b 1 ), sub-base flow (c 1 ) and base flow (d 1 ) are parameters of the Tank model that can be used to describe how moves horizontally in a given catchment. The regionalized models for sub-surface flow (a1), surface flow (a2), intermediate flow (b1), sub-base flow (c1) and base flow (d1) (Table 6) suggest that mean perimeter-area ratio of the broad-leaved forest patches and needle-leaved forest patches as well as mean shape index of the broad-leaved forest patches and mean patch size of the water bodies are directly related to the aforementioned parameters in the catchments.

Although an inverse relationship between the patch density of the needle-leaved forest patches and sub-surface flow (a 1 ) has been observed, patch density of the broad-leaved forest plays a role in explaining the variations in the intermediate flow (b 1 ) and base flow (d 1 ) of the catchments. Variation of the intermediate flow (b 1 ) can directly be explained by change in the patch density of the broad-leaved forest patches, however the latter has been found to be indirectly related to the variations in base flow (d 1 ) of the catchments.

3.3 Water Storage Level- Reflecting Parameters

Water storage level-reflecting parameters include: Ha 1 , Ha 2 , Hb 1 , Hc 1 and H d (Fig. 3), by which water storage levels are indicated in each of the corresponding tanks. These parameters are also known as threshold of flows including surface flow (Ha 2 ), sub-surface flow (Ha 1 ), sub-base flow (Hb 1 ) and base flow (H d ).

Table 6 indicates that the mean perimeter-area ratio of water bodies and agricultural patches are inversely associated with variations in the values of sub-base flow (Hb 1 ). The patch density of grasslands (G pd ) and agricultural patches (A pd ) has two different impacts on the sub-base flow (Hb 1 ). Accordingly, increasing G pd will result in decreasing Hb 1 and vice versa for the later. Moreover, a direct relationship has been observed between the mean shape index and the mean patch size of water bodies with the values of Hb 1 and Ha 2 , respectively. Table 6 also summarizes the statistics of the regression analysis-based regionalization. No inter-variable collinearity of the regression models was detected (VIF < 4.9) (Chatterjee et al. 2000). Observed versus predicted measures of the regionalized models were also plotted to validate the statistically significant models’ goodness-of-fit (Fig. 4).

Fig. 4
figure 4

One-to-one relationship between the observed parameters and calculated parameters of the Tank model

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3.4 Uncertainty Analysis of the Regionalized Models

A Monte Carlo simulation-based uncertainty analysis was performed for the dependent and independent variables of the regression models (Table 7). The cumulative density function, F(x), expresses the probability the random variable X assumes a value less than or equal to some value x, F(x)= Pr (X ≤ x). For this study, the cumulative density function was used to determine the probability of the model output being less than zero (Pr (output) < 0) as only positive values can be assigned to the optimized parameters of the Tank model. The behavior of a given model can therefore be inferred by relying on its cumulative density function. Accordingly, Pr (output) < 0 for a 0 , b 1 , d 1 and Hb 1 are 0.60, 0.90, 0.70 and 0.42, respectively (Fig. 5).

Table 7 Result of uncertainty analysis of dependent and independent variables of the regression models
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Fig. 5
figure 5

Cumulative density function for the simulated outputs of Ln (a 0 ), Ln (a 1 ), b 1 , d 1 , Ha 2 and Hb 1 regression models

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4 Conclusions

The present study was conducted in order to examine a new approach, inspired by landscape ecology, to explain how hydrological processes might occur in a given catchment. In order to examine these possible relationships, a multiple linear regression approach was applied to develop models based on data collected from 30 catchments in Germany. Coefficients of determination for the developed multiple regression models varied between 0.20 and 0.85. It is important to note that the models are restricted to the specific range of variables for the range of catchments used in this study (53 to 737 km2), and that local or regional variations outside of this range could explain the variations of the parameters of the Tank model. Based on the findings of this study, it can be seen that landscape metrics should be used for the regionalization of conceptual rainfall-runoff models.

Relatively robust multiple regression models (0.40 ≤ r 2 ≥ 0.85) were developed for six parameters of the Tank model, including: infiltration coefficient (a 0 ), surface flow coefficient (a 1 ), sub-surface flow (b 1 ), base flow (d 1 ), water storage level for surface flow (Ha 2 ) and sub-base flow (Hb 1 ). The findings suggest that stand alone regionalization of conceptual models should not be a substitute for modeling practices since reliable regionalized models were not achieved for all the parameters of the Tank model. Instead, regionalization could help practitioners to find a more detailed explanation of the hydrological processes in a given catchment.

This study revealed that landscape metrics can explain variations in the model’s optimized parameters. Considering the broad range of landscape metrics, it is highly recommended that further studies be conducted to specify which landscape metrics would be able to explain the variations in the hydrological responses of the catchments.