Geostatistical regionalization of low-ﬂow indices: PSBI and Top-Kriging

Recent studies highlight that geostatistical interpolation, which has been originally developed for the spatial interpolation of point data, can be e ﬀ ectively applied to the problem of regionalization of hydrometric information. This study compares two innovative geostatistical approaches for the prediction of low-ﬂows in ungauged basins. The ﬁrst 5 one, named Physiographic-Space Based Interpolation (PSBI), performs the spatial interpolation of the desired streamﬂow index (e.g., annual streamﬂow, low-ﬂow index, ﬂood quantile, etc.) in the space of catchment descriptors. The second technique, named Topological kriging or Top-Kriging, predicts the variable of interest along river networks taking both the area and nested nature of catchments into account. PSBI 10 and Top-Kriging are applied for the regionalization of Q 355 (i.e., the streamﬂow that is equalled or exceeded 355 days in a year, on average) over a broad geographical region in central Italy, which contains 51 gauged catchments. Both techniques are cross-validated through a leave-one-out procedure at all available gauges and applied to a subregion to produce a continuous estimation of Q 355 along the river network ex- 15 tracted from a 90 m DEM. The results of the study show that Top-Kriging and PSBI present complementary features and have comparable performances (Nash-Sutcli ﬀ e e ﬃ ciencies in cross-validation of 0.89 and 0.83, respectively). Both techniques provide plausible and accurate predictions of Q 355 in ungauged basins and represent promising opportunities for regionalization of low-ﬂows. The comparison is performed at two di ﬀ erent spatial scales: (a) regional scale; (b) catchment scale.


Introduction
Hydrological predictions generally have to deal with the inadequacy or deficiency of observations for the site of interest (see the decade on Prediction in Ungauged Basins promoted by the International Association of Hydrological Sciences -IAHS; Sivapalan et al., 2003). The prediction of low-flows indices in ungauged basins is a fundamen-The regionalization of the low-flow regime is a central topic in this research area (see e.g., Smakthin, 2001;Castellarin et al., 2004;Blöschl, 2006a, Castiglioni et al., 2009), as it can be conveniently applied to surface water assessment in ungauged basins. The scientific literature reports a number of regionalization studies in which the availability of surface water can be evaluated at a given ungauged site on the basis 10 of the streamflow data collected at gauged basins that are hydrologically similar to the site of interest (see e.g., Vogel and Kroll, 1992;Ludwig and Tasker, 1993;Furey et al., 2000).
Recent studies show that geostatistical methods, which have been originally developed for the spatial interpolation of point data (see e.g., De Marsily, 1986; De Marsily 15 and Ahmed, 1987), can be effectively applied for regionalization of hydrometric information (Skøien et al., 2006;Skøien and Blöschl, 2007;Chokmani and Ouarda, 2004). In particular, two different approaches to apply spatial interpolation for prediction of the streamflow regime in ungauged basins have been reported in literature. A first approach, named Topological kriging or Top-Kriging, uses a description of the morphology 20 and nested structure of the study area for spatially interpolating the streamflow index of interest (e.g., flood quantiles, low-flow indices, etc.) along the stream network (Skøien et al., 2006;Skøien and Blöschl, 2007).
A second approach spatially interpolates the hydrometric index of interest in a twodimensional space of physiographic and climatic descriptors, the so-called physio- 25 graphic space (Chokmani and Ouarda, 2004). The x and y coordinates of the physiographic space are derived from an adequate set of n > 1 geomorphoclimatic catchment descriptors through the application of multivariate techniques such as the Principal Component Analysis (PCA) or Canonical Correlation Analysis (Chokmani and Hundecha et al., 2008). In particular, with the Physiographic-Space Based Interpolation, or PSBI, any given basin (gauged or ungauged) can be represented as a point in the x-y space described above; in the same way the set of gauged basins of the study area can be represented by a cloud of points in this space. The empirical values of the quantity of interest (e.g., flood quantile, low-flow index, long-term annual 5 streamflow, etc.) can be represented along the third dimension z for each gauged catchment, and can then be spatially interpolated by applying a standard interpolation algorithm. The spatial interpolation enables one to represent the quantity of interest over the entire portion of the x-y space containing empirical data, and therefore to estimate it at ungauged sites lying within the same portion of the space. The literature 10 reports successful applications of PSBI to the problem of regionalization of flood frequency regime (Chokmani and Ouarda, 2004) or parameters of rainfall-runoff models (Hundecha et al., 2008) and low-flows (Castiglioni et al., 2009).
Although rather different in their approach to regionalization, Top-Kriging is intimately connected to the stream network structure and geometric layout (i.e., Top-Kriging is 15 only dependent on catchment boundaries), while PSBI relies on the identification of an orthogonal space to represent hydrological proximity, these methodologies share a common background idea. Both methodologies perform the regionalization of streamflow indices seamlessly over the stream network (Top-Kriging) or the physiographic space (PSBI) without defining or identifying homogeneous regions or pooling-groups 20 of sites (see e.g., Burn, 1990;Reed et al., 1999;Castellarin et al., 2001). The approaches avoid the phase of catchment grouping, or catchment classification, a challenging task which is generally performed through procedures that are associated with some elements of subjectivity, and whose practical usefulness for predictions in ungauged basins is still an open and highly debated hydrological problem (see e.g., Mc-25 Donnell and Woods, 2004).
We present in this study a comparison of Top-Kriging and PSBI aimed at assessing their ability in predicting low-flow indices in ungauged sites. We applied Top-Kriging and PSBI to a geographical region in central Italy.  Castiglioni et al., 2009). The study focused on the prediction of Q 355 , the daily streamflow associated with duration equal to 355 days, or equivalently the daily streamflow that, on average, is equalled or exceeded 355 days in a year. Q 355 is commonly used 5 in Italy as a low-flow index, and Castiglioni et al. (2009) showed for the study area that this low-flow index is highly correlated to two other commonly adopted low-flow indices, the daily discharge equalled or exceeded 95% of the time, Q 95 , and the 7-day discharge with a recurrence interval of 10 years, 7Q10 (also see e.g., Smakhtin, 2001;Laaha and Blöschl, 2006b;Gustard et al., 1992;Telis, 1992).

10
The comparison is performed at two different spatial scales: (a) regional scale; (b) catchment scale.
1. We assess the ability of each technique to predict Q 355 through a comprehensive leave-one-out cross validation procedure for the entire study region.
2. We then apply both methodologies to a large catchment of the study area to 15 better analyse and interpret their accuracy and reliability for predicting Q 355 along the stream network.
The analysis enables us to assess the performances of Top-Kriging and PSBI and to compare advantages and disadvantages associated with the application of each methodology, gaining additional insights on the applicability and usefulness of geosta-20 tistical techniques for the prediction of streamflow indices in the PUB context.

Geostatistical regionalization
Geostatistical regionalization is herein used to spatially interpolate low flows, by applying kriging over a two-dimensional space of catchment descriptors (PSBI) and techniques developed to interpolate spatially autocorrelated variables, where the spatial coordinates may either identify geographical location (this is the most common case) or a position in a generic bidimensional space. In particular kriging is a method for optimizing the estimation of a quantity that is distributed in space and measured at a network of points (see e.g., Journel and Huijbregts, 1978;De Marsily, 1984;Isaaks and 5 Srivastava, 1989;Rossi et al., 1992;Chokmani and Ouarda, 2004). In kriging, the spatial interpolation is obtained by a linear combination of the observed values according to the following equation: . . N, and λ i is a weighting coefficient. These weights are estimated by considering spatial correlation and configuration of the observations through variogram models fitted to experimental variograms. An experimental variogram expresses the semivariance between observations as a function of distance and direction of pairs of sampling locations, and describes the spatial cor- 15 relation structure of the sample data (see e.g., Journel and Huijbregts, 1978;Cressie, 1993). In practical applications, theoretical variogram models are fitted to experimental variograms to ensure a positive-definite covariance matrix. The literature proposes several theoretical variograms (linear, spherical, gaussian, exponential variogram, etc., see for 20 instance Cressie, 1993;Journel and Huijbregts, 1978).

PSBI: Physiographic Space-Based Interpolation
The physiographic space-based interpolation applies spatial interpolation techniques, such as kriging, to the physiographical space defined by the catchment descriptors of the selected group of basins. The space is defined using two orthogonal coordinates  (Chokmani and Ouarda, 2004). One of them is Principal Component Analysis technique (PCA).

5
PCA is an exploratory multivariate statistical method which is often used to reduce the dimensionality of complex datasets. PCA allows to transform a n-dimensional space into a new space defined by m variables (namely principal components), with m between 1 and n, which are linear combinations of the original variables. The principal components are uncorrelated and orthogonal to one another. They are ordered in such 10 a way that the first component explains the largest amount of variance in the original dataset; the second explains the largest remaining amount of variance, and so on. When reducing the dimensionality of a given dataset, the selection of the appropriate number of principal components can be based upon one of the criteria reported in the literature (i.e. explained variance, Kaiser criterion, scree diagram test; Jackson, 1991; 15 Cattell, 1966). Therefore PCA enables one to present most of the information present in the original dataset in the first m principal components. In particular, the first and second principal components, which are uncorrelated and orthogonal, can be used to define a two-dimensional space for applications of spatial interpolation techniques (Chokmani and Ouarda, 2004;Castiglioni et al., 2009). 20

Top-Kriging
The main idea of Top-Kriging (Skøien et al., 2006, Skøien and is to combine two groups of hydrological variability. The first group consists of variables that are continuous in space such as rainfall, evapotranspiration and soil characteristics, which are related to local runoff generation. In Top-Kriging, the variability of these continuous 25 processes in space is represented by the variogram. The second group of variables, such as runoff, is related to routing in the stream network. These variables are only defined for points on the stream network. In Top-Kriging the aggregation effects that lead 7237 Introduction to these variables are represented by the catchment boundaries associated with each point on the stream network. Rather than using variograms directly, Top-Kriging uses point variograms averaged over the catchment areas. These averaged variograms depend on the point variogram as well as the sizes and the relative positions of the two catchments that are compared by the variogram. In a first step of the analysis, a point 5 variogram model needs to be estimated from the data. This is done by estimating a sample variogram from the data, not only based on the centre-to-centre distance between catchments, but also on their size. From this, the point variogram model can be back-calculated by fitting aggregated variogram values to the sample variogram. In a second step, the point variogram is aggregated to the sizes and positions of the 10 catchments for which the hydrometric index of interest is to be estimated and kriging is performed to predict the variables at the ungauged locations (Skøien et al., 2006;Skøien and Blöschl, 2007). Top-Kriging provides both the estimates as well as the kriging variance (uncertainty). 15 The study area includes 51 gauged basins (∼2.8 gauges per 1000 km 2 ) for which daily streamflows are available for different observation periods with a minimum record length of 5 years (see Fig. 1 and Castiglioni et al., 2009 and minimum values. These values show that the study area presents some degrees of heterogeneity from a hydrological point of view, the major sources of variability being the sizes of the considered basins, their permeability and altimetric characteristics, and mean annual precipitation; an analysis of temperature regime at catchment scale (i.e., mean annual and monthly temperatures) revealed a limited variability (see also 5 Castellarin et al., 2004 andCastiglioni et al., 2009). Concerning the catchment scale application, our study focuses on the Metauro catchment (see Fig. 1). The Metauro's basin morphology is analyzed on the basis of the SRTM 90 m Digital Elevation Model, DEM (see Consortium for Spatial Information, CGIAR-CSI, www.cgiar-csi.org), which was produced by NASA during the NASA 10 Shuttle Radar Topographic Mission (SRTM). The soil properties of Metauro's basin are retrieved from a pedo-lithological map of Marche and Abruzzo (see e.g., Centamore et al., 1991;Vezzani and Ghisetti, 1998).

Structure of the analysis and results
4.1 Regional-scale analysis 15 4.1.1 PSBI PSBI was applied by using universal kriging. This geostatistical interpolator was selected on the basis of previous analysis performed over the same geographical area, which showed the superior performance of the kriging interpolator relative to deterministic interpolators for the application of PSBI to the prediction of Q 355 in ungauged 20 basins (see Castiglioni et al., 2009).
Universal kriging was performed over the physiographic space defined by two orthogonal coordinates. The coordinates were identified from the set of nine geomorphoclimatic descriptors illustrated in Table 1 by performing a Principal Components Analysis (PCA, Basilevsky, 1994;Chokmani and Ouarda, 2004;Castiglioni et al., 2009 linear combination coefficients for the first two principal components are depicted in Fig. 2. The first two principal components present eigenvalues greater than one, which is required by the Kaiser criterion, and explain together about 70% of the variability of the original set of physiographic and climatic descriptors. Several theoretical variogram models reported by the literature (i.e., exponential, gaussian, bessel, circular, spherical, pentaspherical) were tested in a series of preliminary analyses and the best predictions of the quantity of interest were obtained with the spherical model. The three parameters of the variogram model were fitted through a Weigthed Least Squares (WLS) method of the routine fit.variogram of the R-package Gstat (Pebesma, 2004). Figure 3 reports empirical and theoretical variograms for the 10 the study region.

Top-Kriging
Regional application of Top-Kriging for the entire study area utilises as input data watershed boundaries for the 51 considered basins. These boundaries have been previously identified on the basis of a set of 1:100 000 topographic maps. This information is used 15 in Top-Kriging for reproducing the nested structure of the catchments, and for carrying out the discretization of study basins into square cells, and for calculating distances between cells of the grid for the discretized catchments (Skøien et al., 2006). A modified exponential variogram is adopted to model the empirical variograms; the model is fitted to sample data through a WLS method implemented in the R-package 20 Rtop (Skøien, 2009). Figure 3 illustrates the point variogram and regularised variograms of different catchment sizes, and a comparison between binned empirical and modelled semivariances. Skøien and Blöschl (2007) focus on the specific, that is per unit area, runoff q(x i ) at location x i along the stream network. Therefore we applied Top-Kriging by referring to

Results
We assessed the reliability of the techniques and the uncertainty of the associated predictions of Q 355 in ungauged basins by applying a leave-one-out cross-validation procedure; this procedure often referred to as jack-knife in the literature and in the remainder of this study (see e.g., Efron, 1982;Zhang and Kroll, 2007;Brath et al., 2003;5 Castellarin et al., 2004;Castiglioni et al., 2009). This method is extremely versatile and evaluates the performance of interpolation techniques by simulating the ungauged conditions for each and every one of the 51 basins in the study area. In the procedure we assume one of the N basins, let us say site i , to be ungauged, and predict the Q 355 value for this site on the basis of the remaining N − 1 observations. Repeating 10 this step N times, considering in turn each of the basins as ungauged, we obtain Ncross-validation estimates of Q 355 which can be compared with the corresponding observations. In particular, for the cross-validation of PSBI we did not update the Principal Components since they are associated with catchment descriptors that can be derived also for ungauged sites (i.e. for site i ), but we did update the variogram 15 Ntimes, each time discarding the streamflow information for site i and refitting the model to the available sample data. The scatterplots of empirical vs. estimated values of Q 355 resulting from the applications of PSBI and Top-Kriging in cross-validation are shown in Fig. 4. Jack-knife predictions and empirical values of Q 355 were quantitatively compared in 20 terms of Nash-Sutcliffe efficiency criterion, E (Nash and Sutcliffe, 1970) and mean absolute relative error, ε rel . In statistical modelling of flow indices, the term "coefficient of determination R 2 " is often used instead of E . We prefer to refer to E to avoid possible ambiguities with the squared linear correlation coefficient, another commonly used performance index that, differently from E , cannot detect a bias in the predictions. Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | regional dataset of the three basins for which Top-Kriging gives large relative errors in cross-validation (red dots, Fig. 4). These gauges are all characterized by very low values of specific low flow, q 355 . These catchments present q 355 values smaller than 0.15 l/s/km 2 , whereas the average value of q 355 for entire study area is 5.13 l/s/km 2 . For two of the three catchments the very small q 355 values derive from their atypical 5 geomorphological characteristics: low permeability (P values are equal ∼0.1% and 8%, and are among the smallest for the entire study region), closeness of the catchment boundaries to the shoreline and limited mean altitudes. The third catchment is a very small headwater catchment, characterized by high elevations and karst phenomena. Furthermore, these basins are geographically far apart from one another, 10 and the neighbouring catchments are characterised by definitely higher q 355 values. All these circumstances limit the applicability of Top-Kriging, whose estimates depend on catchments' boundaries only and result in significant overestimations of q 355 values (see Fig. 4), which results in remarkable relative errors due to the small q 355 empirical values. Therefore, we find it useful and fair to compare the performance of PSBI and 15 Top-Kriging indices with and without the three particular catchments.
The results indicate that, if we exclude the three peculiar basins, PSBI and Top-Kriging have similar performances. This evidence is supported by the comparable values of the performance indices reported in Table 2. The performances of the two methodologies are similar in terms of E with or without the three peculiar catchments, 20 and, as a results of the considerations stated above, get similar also in terms of relative errors when the three catchments are neglected.

Catchment-scale analysis
To further advance the comparison, Top-Kriging and PSBI were also implemented at the catchment scale to predict Q 355 in the Metauro basin (see Fig. 1). The Metauro Top-Kriging, only the streamflow data observed at the 7 stream gauges were considered (i.e., gauge 1002 together with 6 internal gauges). A first analysis applied PSBI and Top-Kriging to interpolate Q 355 along Metauro's stream network (see Sect. 4.2.1). A second analysis applied the two methodologies in cross-validation and compared empirical and jack-knife predictions of Q 355 for the 7 5 considered stream gauges (see Sect. 4.2.2). Also in this case, only the hydrometric information observed at the 7 stream gauges illustrated in Fig. 5 was considered.

Preparation of the dataset
The catchment-scale application of both methodologies required the preliminary identification of the stream network, the delineation of a subdivision of the Metauro's into 10 subcatchments on the basis of the confluences and network's layout (application of Top-Kriging), and the evaluation of the nine considered geomorphologic and climatic descriptors (see Table 1) for all identified subcatchments (application of PSBI). To this end, we carried out a process characterized by four steps (see Fig. 6). This analysis allowed us to highlight the advantages and disadvantages of each methodology, em-15 phasizing in particular the straightforwardness/complexity of all steps required for the preparation of the dataset required by the implementation of both techniques. This is a critical matter in the context of Prediction in Ungauged Basins (PUB).
Below we describe the four steps exemplified in Fig. 6, which we carried out in a GIS (Geographic Information System) environment: 20 1. Stream network and basin contours were identified from the 90 m DEM (GIS processing 1 in Fig. 6). First we identified the stream network by means of the eightdirection-pour-point (or D8) algorithm (see e.g. Brath et al., 2002) on the basis of a suitable critical source area (CSA). CSA was identified by reaching an acceptable level of congruence between the river network retrieved from the DEM and 25 that reported on 1:25 000 topographic maps. Afterwards, all subcatchments with drainage area ≥ 10 km 2 were identified for the Metauro catchment, which resulted into 44 additional ungauged sub-basins (see Fig. 5).

2.
A,L,H max , H mean , H min, τ c and ∆H were computed for each subcatchment from the 90 m DEM (GIS processing 2 in Fig. 6). The physiographic descriptors computed from the DEM were compared with the data reported in the literature (see Rep. No. 17; Min. LLPP., Italian Dept. of Public Works, various years) for the 5 drainage areas of seven stream gauges belonging to the Metauro's catchment. All parameters retrieved from the DEM were found to be practically coincident with the literature values.
3. By combining catchment boundaries with a pedo-lithological map, the percentage of permeable area (P ) for each sub-basin was determined (GIS processing 3 in Fig. 6). Also in this case the congruence with the data reported in the literature was verified.

Mean Annual Precipitation, MAP, at catchment-scale was computed for all
Metauro's sub-basins by using the precipitation data observed at several raingauges (see Fig. 6) and the Thiessen polygon technique (GIS processing 4 in 15 Fig. 6).
Principal Components (PC's) are used when applying PSBI at catchment scale to fully characterise the physiographic space for the entire Metauro catchment. Therefore PC's of the catchment scale application were computed using the catchment descriptors derived as described above for a total of 51 catchments, that is the Metauro catch-20 ment, the six internal gauged sub-basin and the 44 additional ungauged sub-basins identified at step 1. Figure 6 immediately points out that the preparation of the dataset for Top-Kriging requires less steps than the preparation of the dataset for PSBI. Nevertheless, it is worth noting here that the part of the procedure that required a lot of "manual" input and 25 that resulted to be the least straightforward was the identification of stream network and watershed boundaries (step 1). This results from the complex morphology of the study area and approximations associated with the adopted DEM. It is also worth remarking that this step is required for the application of both techniques: the application of Top-Kriging requires the basin contours to be known in order to define size and nested structure of all catchments; the application of PSBI requires geomorphological and climatic descriptors computed or retrieved for each sub-basin. 5 As a commentary for the application in the PUB context of both methodologies, one should note that high-to-medium resolution DEM's for the entire world are accessible the on internet (see e.g. www.cgiar-csi.org). Also, referring in particular to the application of PSBI, it is worth remarking here that raingauge networks are generally quite dense in many regions of the world. For these reasons, the identification of all geomor-10 phological and climatic parameters adopted in this study, except for P , can be easily performed in other geographical contexts. The percentage of permeable area, P , is a highly descriptive parameter for our study area (see e.g., Castellarin et al., 2004;Castiglioni et al., 2009), yet complex to determine and to quantify, even though, as showed here, pedo-lithological maps can be used to this end.

Results
We applied Top-Kriging and PSBI for predicting Q 355 along Metauro's river network. A previous application of Top-Kriging showed that this technique provides plausible and accurate predictions of streamflow-related variables along the stream network (e.g., specific 100-year flood, see Skøien et al., 2006, and  We summarize in Table 3 the absolute relative error obtained at each stream gauge (cross-validation). The values reported in Table 3  (Metauro at Barco di Bellaguardia). Both gauges are situated at a river junction and the low flow regime at the confluent river at gauge 1004 vastly differs from the regime of the main river at 1002. This situation contradicts the assumption of intrinsic stationarity of Top-Kriging, i.e. similar low flows at close-by sites. As shown in previous studies, Top-Kriging predicts even well in such a situation, if the heterogeneity is represented by 15 the upstream gauge (Laaha et al., 2010), but performs rather poorly if no information about the heterogeneity is available. This situation is simulated by the cross-validation approach.
It is interesting to observe that the catchment scale application of PSBI and Top-Kriging resulted in balancing performances. The average value of the minima of the 20 relative errors in absolute value computed for the two methodologies is as low as 18% in cross-validation, which is definitely smaller than the mean values for the two methodologies (see Table 3). This is an important result that underlines how different the methodologies are, notwithstanding the common background philosophy (application of geostatistics to regionalization of streamflow regime), and how low-flow predictions 25 in ungauged sites could in principle benefit from a simultaneous application of both methodologies. An agreement between the predictions resulting from the application of the two methodologies supports a good reliability of the predictions themselves, whereas a discordancy for a particular portion of the study area should suggest to further investigate that area, perhaps by planning ad-hoc streamflow measurement campaigns.

Discussion and conclusions
This study aims at comparing two innovative regionalization approaches, namely Physiographic-Space Based Interpolation, or PSBI, and Top-Kriging, for the predic-5 tion of low-flows indices in ungauged basins. Both methodologies rely heavily on the use of geostatistical tools for the regionalization of the streamflow information, even though they adopt two markedly different approaches. A quantitative comparison of the two methodologies has not been reported in the literature yet.
We focused on geostatistical approaches only (i.e., PSBI and Top-Kriging) because 10 a previous analysis performed for the same study area showed that PSBI enables one to deal with heterogeneous regions more effectively than traditional multiregression models, providing more accurate predictions of low-flow indices in ungauged basins (Castiglioni et al., 2009 information from gauged information of similar catchments. PBSI, however, performs poorer at larger rivers. PBSI, like any multivariate technique, assumes a clear relationship between the low flow index and catchment processes represented by catchment characteristics. The aggregation of discharges at larger rivers leads to a mixing of different regimes and low flow generating processes and therefore reduces the perfor-15 mance of multivariate techniques (Laaha and Blöschl, 2007;Laaha et al., 2010). In this case, Top-Kriging proved more suitable, as it does not rely on such relationships. The fact that the Metauro basin is dominated by headwater catchments gives rise to a somewhat better performance of PBSI than Top-Kriging in this region. Top-Kriging and PSBI are both geostatistical interpolation methods. The main dif-20 ference is that Top-Kriging interpolates low flows in the geographic space, using information from the closest gauges at the river network, while PSBI interpolates low flows in a space of physiographic catchment characteristics, using information from the most similar gauges in terms of catchment characteristics. Top-Kriging is solely based on spatial information and, therefore, requires that spatial heterogeneity of low flows is 25 covered by the location of data points. Heterogeneity not covered by local data will yield higher predictive uncertainty. This is a major difference to multivariate techniques or PSBI, which can transpose such information without relying on local data. Multivariate techniques, however, require that spatial heterogeneity of low flows can be explained by the available catchment characteristics and the model. The fact that relevant catchment characteristics, such as storage coefficients or permeability, are to a large extent unknown, impedes the effectiveness of multivariate statistical techniques. From these findings, we can conclude a complementariness of the two estimation methods. The methods are complementary in terms of the basic principle of spatial 5 interpolation, because Top-Kriging is based on proximity in geographic space while PSBI is based on proximity in terms of catchment characteristics. The methods are also complementary in terms of data requirements, since Top-Kriging requires that heterogeneity of low flows is covered by the gauging network, while PSBI requires that heterogeneity to be covered by catchment characteristics and the model. Finally, the 10 methods are complementary in terms of predictive performances, since Top-Kriging yielded better results for homogeneous situations along the main rivers while PSBI yields better results for heterogeneous situation in headwater catchments. These different properties of the methods are important, especially for practical applications when predicting low flows of ungauged basins, where the choice of the method shall 15 optimally be guided by available data and the hydrological situation. The complementariness of the methods also suggests a combination of the methods, to make best use of the merits of both techniques.
This study represents a preliminary effort, further analyses in different geographical and climatic contexts are needed for a more comprehensive assessment of these 20 methodologies, their effectiveness for predicting low-flow in ungauged basins, and their degree of complementariness.