A distributed or semi-distributed
deterministic hydrological model should consider the hydrologically most
relevant catchment characteristics. These are heterogeneously distributed
within a watershed but often interrelated and subject to a certain spatial
organization which results in archetypes of combined characteristics. In
order to reproduce the natural rainfall–runoff response the reduction of
variance of catchment properties as well as the incorporation of the spatial
organization of the catchment are desirable. In this study the width-function
approach is utilized as a basic characteristic to analyse the succession of
catchment characteristics. By applying this technique we were able to assess
the context of catchment properties like soil or topology along the
streamflow length and the network geomorphology, giving indications of the
spatial organization of a catchment. Moreover, this information and this
technique have been implemented in an algorithm for automated sub-basin
ascertainment, which included the definition of zones within the newly
defined sub-basins. The objective was to provide sub-basins that were less
heterogeneous than common separation schemes. The algorithm was applied to
two parameters characterizing the topology and soil of four mid-European
watersheds. Resulting partitions indicated a wide range of applicability for
the method and the algorithm. Additionally, the intersection of derived zones
for different catchment characteristics could give insights into sub-basin
similarities. Finally, a HBV

Hydrological models are instruments for structuring the knowledge of
hydrological processes in their dependence on watershed characteristics. For
the set-up of these models several initial decisions have to be made, e.g.
the following.

Which type of model has to be used?

Which temporal resolution could be appropriate?

Which spatial resolution of the model would be necessary and useful?

Which way could the model be parameterized?

It is obvious that all of these options will affect the effort of the model and all choices have to consider the modelling purpose. Furthermore, these choices are interrelated; for example, predominant soil properties define the dominant runoff process and should, hence, define the used model. Therefore, conceptual models of a natural watershed require its subdivision into spatial units which should be as homogeneous as possible. Hydrological modelling is the attempt to specify hydrological processes quantitatively under consideration of boundary conditions. These boundary conditions are mainly determined by spatially heterogeneously distributed catchment characteristics. There are several approaches to address this heterogeneity in models to enable work with more or less homogeneous units.

One option to address spatial heterogeneity might be the subdivision of a river basin into sub-basins which have to be modelled separately. The common approach for such a subdivision is usually based on available hydro-meteorological data, though the correct criteria would be the spatial heterogeneity of hydrological characteristics within the river basin. If the heterogeneity is at a low level, neighbouring basins could be modelled in accordance. In the reverse case, i.e. at a high level of heterogeneity, sub-basins should be modelled separately with an especially adapted model taking into account their specific characteristics (e.g. an urban watershed model). Subsequently each sub-basin needs to be treated as a unique modelling instance that should provide a minimum level of heterogeneity (regarding key catchment characteristics). This way each sub-basin would end up with its own unique model and/or parameter set to adjust the model to mimic its natural response.

Another option to address spatial heterogeneity within a watershed could be to split the catchment into so-called hydrologic response units (HRUs). A single unit merges areas, or cells, within a basin displaying similar characteristics independent of their respective spatial allocation, i.e. each unit is a unique modelling instance. The HRU approach is based on the key assumption that the variation of the hydrological process dynamics within the HRU must be low relative to the dynamics in another HRU (Flügel, 1995). HRUs are developed by intersecting different data layers of different physiographic criteria. The delineation of HRUs by combinations of these layers requires a categorization of its characteristics (soils, land-use and vegetation types, topography, and geology) to keep the number of HRUs at a manageable level, both the selection of criteria and their subdivision into classes at an acceptable degree of heterogeneity within the hydrological system. Subject to the chosen technique or purpose of HRUs, their models omit the actual spatial allocation (Lindström et al., 1997; Schumann et al., 2000) or define coherent units (Dunn and Lilly, 2001; Soulsby et al., 2006; Müller et al., 2009; Nobre et al., 2011; Gharari et al., 2011). However, some of these models try to transfer geological information (Müller et al., 2009; Soulsby et al., 2006) or topographical information (Nobre et al., 2011; Gharari et al., 2011) to hydrological processes and assume homogeneous conditions of the remaining parameters.

A third option to address spatial heterogeneity in hydrological modelling is the utilization of a distributed catchment characteristic as a covariant metric supporting the spatial distribution of a lumped state variable. An example of this approach is the use of the topographic index in the well-known TOP model as a characteristic of the spatial variability of the soil water content (Beven et al., 1984).

Since GIS layers are widely available, there is an obvious trend to incorporate these data into the ascertainment of spatial units. Most approaches are based on topography (Band, 1986; Moore and Grayson, 1991; Vogt et al., 2003; Lai et al., 2016) and focus on the extraction of stream networks and the network connectivity, utilizing topology-driven modelling concepts (Beven and Kirkby, 1979; Rodríguez-Iturbe and Valdés, 1979). Particularly the development of the geomorphologic instantaneous unit hydrograph (GIUH) as well as its enhancements like the geomorphological dispersion (Rinaldo et al., 1991; Gupta and Mesa, 1988) require sophisticated stream network derivation and analysis. Methods introduced by Band (1986) or Verdin and Verdin (1999) were developed to generate data to meet this requirement. Furthermore, Snell and Sivapalan (1994) applied the width function, introduced by Kirkby (1976), to model the geomorphic structure of networks. Snell and Sivapalan (1994) were able to demonstrate that GIUHs based on the width function provide a better geomorphic dispersion than GUIHs derived from Horton laws (Robinson et al., 1995; D'Odorico and Rigon, 2003; Rigon et al., 2016). This has been a further step to incorporate remote sensed data in describing the organization of catchments. While the above-described methods are based on gridded digital elevation models (DEMs), other methods try to identify streamlines derived from DEM shapes, producing contour lines (Moore and Grayson, 1991; Lai et al., 2016) that are subsequently used as modelling instances.

The following sections of this paper will present a combination of different methods to address spatial heterogeneity of watershed characteristics, utilizing patterns resulting from the spatial organization of catchments. Sivapalan (2005) pointed out that the organization of a catchment has a fundamental influence on the hydrological system. He defined the organization of a catchment as patterns of symmetry between soil, topography, and the stream network. These patterns could unveil underlying mechanisms that induce discharge behaviour. Combining soil data with the flow path lengths at hillslopes in particular could provide a better understanding of lateral flow distribution processes (Grayson and Blöschl, 2001).

In this study we will present a method to address these patterns by combining the width function (Kirkby, 1976; Mesa and Mifflin, 1986) with soil properties like pore volume and topographic characteristics like surface slope. Unlike traditional methods for spatial pattern evaluation (like point-by-point or optimal logical alignment methods; Grayson and Blöschl, 2001) we retain the allocation of catchment properties. This analysis revealed the organization of the watershed and gave indications of spatial heterogeneity gradations which could be useful for the set-up of an appropriate model structure.

Applying this method we developed an algorithm for automated sub-basin ascertainment. Our objective was to incorporate the spatial organization of watersheds into the spatial structuring of a semi-distributed model and to assess its benefits for model performance. The purpose of the proposed algorithm was to provide a basin partition with a minimum of heterogeneity by a minimum of sub-divisions, i.e. to reduce the number of unnecessary sub-divisions and subsequently the number of parameters in cases of hydrological modelling.

The proposed algorithm has been applied in a case study to four meso-scale
mid-European watersheds and in a HBV

The presentation has been split into four sections.

Data, giving references to observations in some of the basins during the description of the methodological development.

Methodological development, including observations, considerations, and techniques to assess spatial patterns of catchment characteristics and their spatial organization. In this section the sequence of the proposed algorithms will be presented which was utilized to incorporate the spatial organizations into the model.

Method analysis, i.e. checking their applicability and their limits.

Method application, including the subdivision case study and modelling application.

Due to the fact that the proposed methods were based on GIS-based catchment
analysis, we first had to establish our database. In general, four catchments
were selected to develop and test our methodological approach, with one
catchment serving as a development catchment and the remaining catchments for
validation. Our development catchment was the Mulde River basin (Fig. 1,
left). The basin is located mainly in eastern Germany, with a small section in the
northern Czech Republic. Its southern section is located in the mid-range
mountainous region of the Ore Mountains. With a size of 6170 km

Digital elevation models of the Mulde

A DEM is essential for the proposed methods and the algorithm. We used a gridded DEM derived from the Shuttle Radar Topography Mission (SRTM) with a regular resolution of 100 m. By application of the D8 algorithm the required data, i.e. flow directions, flow length, and flow accumulation (i.e. the number of cells draining to the respective cell), were calculated (Jenson and Domingue, 1988). For the catchment of the Mulde River a proved digital river network was already available. Stream networks of the remaining basins were calculated via flow accumulation algorithms. To characterize the soil characteristics of the German catchments, a gridded soil data map from the German Federal Institute for Geosciences and Natural Resources (BÜK200) and CORINE land coverage data (CLC) (Bossard et al., 2000) were used. Pedo-transfer functions (Sponagel, 2005) were applied to transfer this information into gridded data of (available) water capacities (AWCs), maximum soil storage capacity (referred to as total pore volume – TPV), and hydraulic conductivity (HC) for the upper soil, up to 2 m in depth. In the case of the Salzach basin precast pore volume data provided for Europe along with the LARSIM-ME model were used, due to a lack of soil data (Bremicker, 2016). The used soil and topography input data unfortunately include a certain level of uncertainty because these were derived data. However, this was assessed to be negligible for the performed case studies. The pore volume (TPV) data were summarized in Fig. 2.

Values of total pore volume of the Mulde

This section introduces an algorithm that characterizes the heterogeneity of regions and applies techniques to efficiently subdivide the watershed. In order to make the sequence of the algorithm easier to understand, we will first introduce all required new techniques and methods and then present the algorithm.

First, the underlying approach of the distance-factor function for the assessment of spatial organizations (Sect. 3.1) will be presented. Next the tools of the algorithm (Sect. 3.2) will be introduced, followed by Sect. 3.3 presenting the sequence of the proposed algorithm. All tools are only introduced briefly; a detailed description of the applied methodologies can be found in the Appendix.

Throughout the development and ongoing research of the geomorphologic instantaneous unit hydrograph (Rodríguez-Iturbe and Valdés, 1979; Gupta et al., 1980), the width function as introduced by Kirkby (1976) and the subsequently developed area function or weight function (Mesa and Mifflin, 1986; Snell and Sivapalan, 1994; Robinson et al., 1995) were applied to describe the distribution of runoff-producing area with respect to flow distance from the outlet (Mesa and Mifflin, 1986). In particular, the weight function provided the probability distribution for a uniform areal precipitation intensity for the choice of flow path (Snell and Sivapalan, 1994). Since the flow path and distance are known, we can describe the hydrograph at the outlet of a basin under the assumption of a uniform velocity.

However, velocity in reality is not uniform in a basin. It is subject to its surrounding medium (soil, air, other water particles) and the medium condition (dry or wet/empty or full) and a wide range of other impact factors. We would describe the transformation from the arrival of precipitation at its flow path to the outlet of the catchment as the trail function. It merges losses and retention of water along the flow path. The detailed description of the trail function is part of a hydrological model which, at this point, is open to the deliberate choice of the user.

In general nearly all hydrological models require information about catchment characteristic(s) or at least homogeneous conditions of a single characteristic (in most cases, soil properties). Coming back to the idea of the weight function, it seemed worthwhile to develop a method to assess an arbitrary catchment characteristic in the same manner.

We propose the

To assess groups of hillslopes and to account for the non-continuousness of
grid-based distances, we substituted the estimated flow lengths
(

Let us now look at a single distance class

Exemplary input data

Distance-factor function of sample data in a synthetic catchment.

Distance-factor function of AWC in the Mulde catchment.

Figure 5 shows the application to real data in a meso-scale catchment, namely
for AWC in the Mulde catchment. Expected values and a 1-

Taking this approach we will be able to assess the arrangement of catchment properties with the flow path which will, in the case of a non-random arrangement, be referred to as the spatial organization of the basin. Please note that we will always co-notate which type of distance-factor function (expected value or standard deviation) has been applied.

The example of a distance-factor function in Fig. 5 shows that in some distance classes the AWC values are at a similar level due to a low standard deviation. This could be caused by the small size of the distance class (which is the case close to the outlet and in the furthest distance class), but also by the fact that the class is located in the same region (low lands, mountainous regions). The example also shows that distance classes between 50 and 170 km are more heterogeneous, recognizing its higher standard deviations. In order to minimize heterogeneity in these regions, a further subdivision is required in these parts of the catchment.

To ensure more homogeneous sub-basins, an algorithm was developed based on
consisting of the following.

An objective function which identifies the needs and (if necessary) the region of further subdivisions.

A tool to specify the subdivision points in the selected region for subdivision of the catchment.

An evaluation strategy to assess performed subdivisions.

The following subsections give a brief introduction to these three functionalities, before the sequence of the algorithm is described. More details on the introduced tools have been listed in Appendix A.

As outlined before, the standard deviation

Distance-factor function of

As the index

The functionality of the proposed tools will be shown for the synthetic catchment (Fig. 3) introduced in the previous sections (more details in Appendix A). The application of the objective function was expected to result in one of the listed three potential outcomes:

the standard deviations in all distance classes stay below threshold

only parts of the classes have values of

standard deviations of all classes are larger than

The first listed outcome would indicate that no further subdivision is required. The second would indicate that parts of the flow path display nearly homogeneous characteristics. This case is shown in Fig. 7a. Hatched cells indicate the low-variance region. Since this region is homogeneous it will not require any further handling and can be separated from the residual, more heterogeneous parts of the basin.

To do so, a tool called

Both remaining heterogeneous sub-basins, in this example, consist of distance classes with standard deviations above the threshold. This could be caused by different spatial patterns in the sub-areas. On the one hand, parallel streams, or more specific neighbouring valleys with different vegetation, slopes, etc., could cause higher variance. On the other hand, a zoning of hillslopes and higher elevated parts of the basin framing the drainage network (e.g. gley horizons close to streams) could result in higher values of the standard deviations. Such patterns are a result of the co-evolutional formation of catchments (Blöschl et al., 2013; Sivapalan, 2005).

Further tools were developed specifically for these two different potential
root causes of heterogeneity. The first tool will provide a subdivision at
stream branches (because our perspective of analysis is directed upstream;
downstream would mean confluences) which will define new sub-basins at
branching points of higher-order streams. The

The second tool is a

In Fig. 7d and e the results of an application of both tools for the
exemplary synthetic catchment are shown. Hatched cells in the lower left area
indicate high-variance regions that require partition. The

The introduced tool names will be used in Sect. 3.3 (and Fig. 8) where a detailed description and explanation of the algorithms' sequence will be provided.

Sequence of the ACS algorithm.

After the application of each previously introduced tool, it has to be
evaluated as to whether its target has been achieved, i.e. minimizing
heterogeneity through the introduction of sub-basins and zones. Since our
assessment of heterogeneity was based on the evaluation of distance classes,
we could also define our objective as the minimization of the standard
deviation within each distance class. No matter which tool has been applied,
in some or all distance classes of the original sub-basin

As a first step we estimated the standard deviation

This was done for all sub-basins by adding the streamflow length between their points of confluence and the outlet of the basin.

Finally, the new standard deviation

with

The success of the partition can be measured by comparison with
the standard deviation of the unseparated basin

The tools presented above are at this point incoherent. Their sequential application in the ACS (Ascertainment by Catchment Structure) algorithm will be explained step-by-step following the sequence shown in the flowchart in Fig. 8. Our considerations leading to the presented sequence can be summarized as follows.

Homogeneous regions should be separated from the remaining basin by the algorithm.

Preferably a basin should be subdivided into sub-basins at major branches/confluences of the river network.

For high-variance regions the two options

The results of both techniques (

At the very beginning of the sequence, on initialization of the algorithm, we
would consider just one drainage point, i.e. at the outlet of the basin.
After calculation of its watershed and the determination of the width
function of accumulated partial areas, we would evaluate whether we also
needed to consider major branches (see the Appendix for a detailed
description of this procedure). Major branches would account for larger, main
rivers within a catchment. Since large rivers consist of a large number of
cells draining into them, branches/confluences of rivers can be easily
differentiated by size. If the test for major branches turns out to be true,
the

Results of ACS
application for catchments of the Mulde and Regen, sub-basins based on pore
volume

Should a subdivision occur, the algorithm would start again at the previously
used point, this time only looking at the watershed between the previous and
newly defined drainage points. If no further major branch is present, the
algorithm calculates the standard deviations of the characteristic of
interest and the objective function to estimate the number of distance
classes above the threshold

There are three potential outcomes (see Sect. 3.2.2), i.e. standard deviation
in none, some, or all classes above a threshold. For each outcome the
algorithm had an option.

If no standard deviation of a class is above

If only some classes are above

In the case that all classes were above threshold points 3 and 4, our consideration would be started. Both tools would try to lower the heterogeneity assuming different root causes of variation. Results leading to a lower residual variance would be saved; other results would be discarded. The algorithm would then start again at its last active point.

This process would be repeated until all drainage points have been examined by the algorithm. The fact that each basin, or sub-basin, is analysed again after each subdivision provides the opportunity to apply a pre-partition. This could be useful if an existing structure (like a gauging network) is analysed for further improvements or just for zonal classification.

In this section we will analyse the outcome of applying the proposed algorithm for our case study catchments. First, we will take a qualitative look at the ascertained sub-basins and zones to assess similarities between catchments and characteristics. Subsequently we will take a quantitative look at the performance of the algorithm relative to its intended function.

The ACS algorithm has been applied to all four catchments for pore volume and
surface slope. Ascertained sub-basins and zones are shown in Figs. 9a–d
(Mulde and Regen) and Fig. 10a–d (Main and Salzach). Additionally, the
distance-factor functions for standard deviation

Looking at the results it can be noted that the proposed zonal classification was applicable to almost all catchments. Just one sub-basin in the Mulde catchment rejected a zonal classification (Fig. 9a, red sub-basin). Additionally, the defined sub-basins for both characteristics (same catchment) are comparable and often identical. This is mostly caused by the subdivision at a major branch. Nevertheless, differences in the number and extent of defined sub-basins are visible. More important though are the similarities and dissimilarities of the defined zones.

Both applications within the Regen (Fig. 9c, d) and Main (Fig. 10a, b)
catchments resulted in similar patterns of “

Results of ACS application for catchments of the Main and Salzach
(from top to bottom), sub-basins based on pore volume

The same analysis for surface slope zones shows that these zones are in most
cases more extensive than the pore-volume zones and follow the valley
structure of the DEM (Fig. 1). CTS zones cover the streams and floodplains at
the bottom the valleys, “

The interaction of zonal extent is best visible in the Salzach catchment,
which is the most diverse in all of its characteristics (very high mountains
with high slopes, soils with high and nearly no storage capacity). A
comparison of the outcome for pore volume and slope (Fig. 9c, d) with the
respective maps (Figs. 1 and 2) shows that the “

From this analysis of spatial natural patterns and patterns in the ascertained sub-basins and zones, we can draw the conclusion that the outcome of the ACS algorithm is linked to the spatial organization of the considered catchment.

Having confirmed that the proposed algorithm can actually mirror the spatial
organization of the catchment, we will now evaluate whether the heterogeneity
of the specific characteristic has been decreased. As stated in Sect. 2.4 the
intended function of the algorithm was the reduction of the number of
distance classes comprising a standard deviation

Results of applications of ACS. Number of ascertained sub-basins and normalized reduction of standard deviation.

AWC of the Salzach catchment and the distance-factor function of

As already indicated in the distance-factor functions of

Focusing on cases with insufficient variance reduction, we were able to identify some limitations of the algorithm. First we will look at the slope application. The achieved reduction was generally low and the remaining variance was still high, but especially the outcome of the Mulde basin is inferior to all other (slope) applications.

The reason for this inferior performance might be the shape and arrangement of the catchment itself. In contrast to the other basins, the Mulde basin can be described as triangular. Several streams arise from the south of the basin and converge, gradually heading to the north, yielding nearly parallel situated sub-basins with the same spatial organization of heights and slope. As can be seen in the distance-factor function, the variance increases in the upstream direction nearly continuously, is equally distributed, and remains on a (comparably) low level (see the ordinate axis of distance-factor functions for the remaining basins in Figs. 9 and 10). In contrast to that, the remaining catchments offer different spatial patterns. Here, headwater catchments with higher elevations and slope lie within same distance classes as plain catchment parts, offering a higher variance and, hence, a better opportunity for subdivision.

Resampled AWC values for the Mulde and Salzach catchments.

Another inferior case is the pore volume application in the Salzach basin. The shape of the basin as well as the amount of variation (see Fig. 10) cannot be explained as previously. Figure 10 shows the map of the AWC in the Salzach basin on a lower scale and the distance-factor function for standard deviation of AWC. On the map red boxes highlight spots within the basin displaying much higher AWC values than its surrounding areas. These spots are also visible in the distance-factor function. For the separated basin (red line) the peaks are still visible after the separation, although the basic height of the line has been lowered. This observation can only be interpreted such that the occurrences of such soil enclosures are a limiting factor for the reduction of heterogeneity with the ACS algorithm. It does not restrict its applicability, though.

In the previous section we concluded that the shape of the Mulde basin in combination with the present surface slope values could have been the root cause of the noticed decrease in performance. Additionally, the arrangement of pore volume values in the Salzach catchment could have also led to a decrease in performance. These conclusions brought up a fundamental question: was it the value range of the considered characteristics or the spatial arrangement/basin shape that caused the issue? In other words: if we could examine the same basin with another set of values, would the outcome, i.e. the number of sub-basins, zonal extent, and performance criteria change?

The problem is that no basin is like another, and even parts of the basin display different structures and shapes than the entire basin. Therefore, it is unlikely that there is only a single causal factor for the noticed performance decrease.

To overcome this issue we performed what we called a “resampling experiment”. The intention was to examine the same basin shape and spatial arrangement with a different set of values (just like a time series analysis). Therefore a quantile exchange of values has been performed.

Due to their similar sizes, the basins of the Mulde and Salzach have been chosen for resampling. First we took the maps for AWC of both basins (Salzach Fig. 1, Mulde not shown but the spatial organization is analogous to TPV, range of values from 51 to 471 mm) and calculated an empirical distribution function for each basin. Subsequently, the AWC values were replaced with their respective empirical quantile level. Finally, the distribution functions were exchanged (Mulde to Salzach, and vice versa) and the quantile levels were replaced with the exchanged distribution function quantiles. Results are shown in Fig. 12. We repeated this procedure with the DEMs as the source data for surface slopes.

With this, we admit, slightly confusing resampling procedure, we virtually relocated the Mulde basin to a steep alpine environment with diverse soils, while the Salzach basin was equipped with mid-range mountainous heights and more homogeneous soil. That way we were able to assess the same basin shape and spatial arrangement with different (natural reasonable) ranges of parameter values.

The ACS algorithm has been applied to the resampled values of pore volume and
surface slope. Ascertained sub-basins, zones, and distance-factor functions
of the respective standard deviation are shown in Fig. 13; performance values
are tabulated in Table 2. In the case of the Mulde basin the outcome did not
change significantly. Ascertained sub-basins (number and shape) as well as
zonal extent were very similar to the original results; performance values
were stable. It can be noted that the distance-factor function for

Normalized reduction of standard deviation for resampled basins.

Results of ACS
application for resampled catchments of the Mulde and Salzach (from top to
bottom) sub-basins based on resampled pore volume

Application to AWC in the Salzach basin showed a significant change in
performance. While the total reduction decreased, the remaining variance
above the threshold was 20 ppt lower than in the original basin. This is
also visible in the distance-factor function of

In conclusion we can state that the actual spatial arrangement, or more specifically its spatial organization, defines the outcome of the algorithm. Since this was our initial intention, this can be noted as a positive study outcome. On the downside we had to recognize that patterns (in this case soil patterns) that do not follow the co-evolutional structure of a basin (between soil and streams) (Blöschl et al., 2013) cannot be captured satisfactorily by the proposed algorithm. Furthermore, a spatially homogeneous variation structure of catchment characteristics, independent of its actual amount of variation, is also complicated to assess with the ACS algorithm. However, we were able to demonstrate that the proposed algorithm works well for catchment characteristics that offer wide range patterns (like soil properties). (In the Supplement to this article we substantiate this statement by applying the algorithm to hydraulic conductivity data; results are in accordance with results of pore volume.)

In the previous sections we have shown how the algorithm works, what results it produces, and what information about the basins we gained from its application. But we have not yet assessed how useful it is and what benefits it could provide. We will address this topic in the following two sub-sections. In comparison to a common subdivision, we will first evaluate its reduction of variance and second show its benefits for designing a hydrological rainfall–runoff model.

The most common subdivision scheme is based on the available gauging network. On the one hand this is due to calibration requirements and on the other hand it is the only source for a reasonable partition of the catchment. Obviously, existing gauging networks are a result of multiple considerations and requirements, e.g. of water management issues. In some parts of the basin it tends to be denser than required to catch the natural heterogeneity within a river basin, but other hydrological aspects (e.g. scale problems) might not be considered sufficiently in the network design.

Comparing sub-basins defined by hydrological networks by looking at the results of the ACS algorithm might show differences in the number of separation points. Such a comparison might help to provide advice to decision makers on where to locate new gauges for reducing variances. It could also provide information about the usefulness of the ACS algorithm. (Please keep in mind that for the decision maker the usefulness of the algorithm might be limited by the informational value of the specific catchment characteristic for runoff generation processes.)

Two benchmark subdivisions were established.

A subdivision based on the gauging network to be compared to the obtained ACS basins (without zones).

A subdivision based on the gauging network with an additional zonal partition by land cover; based on the suggestions by Lindström et al. (1997), an additional third zone, “Rock/bare soil”, has been introduced to account for Alpine structures in the Salzach basin. This is to be compared to the full outcome of ACS.

Gauging networks and defined land cover zones are shown on the left of Fig. 14. The distance-factor function of standard deviations for pore volume (centre of Fig. 14) and slope (right) are displayed in addition to the results of ACS subdivisions. Performance data were summarized in Table 3.

Subdivisions based on gauging network and zonal classification and
distance-factor functions of

Normalized reduction of standard deviation for sub-basins based on gauging network, ACS basins, and gauges and land cover.

Looking at the distance-factor function for pore volume, it can be recognized
that the red line, representing ACS results, is below the blue
(representing gauges) and green
(representing gauges and land cover) lines for nearly all distances, demonstrating
the advantage of ACS. The ACS advantage is also confirmed by the
performance measures in Table 3. The total reduction

Similarly to Sect. 4, results for slopes provided a different impression and
quality. In the distance-factor functions (Fig. 14) we can see that all lines
are on an equal level and show no clear advantage for any of the partition
strategies. Looking at the performance results in Table 3, we can see that
without zonal classification the gauging network has an advantage for ACS
results. Especially the sub-divisions in the Mulde basin are ineffective.
However, with the introduction of zonal classification, the performance
values are at a more comparable level, with

This result is, again, caused by the fragmented nature of surface slope values. As we have shown before, the gauging catchments in isolation will not capture the heterogeneity of surface slopes, and their performance is subject to a zonal classification that can be described as small-scale distributed and fragmented.

Our model of choice is the HBV

The HBV

Our application case was the Mulde catchment due to good data availability. Daily mean discharge, precipitation sums, temperature means, and sums of potential evapotranspiration time series from 1951 to 2011 are available for 39 gauged sub-basins (discharge data available for 20–39 gauges, dependent on the time window).

Two spatial model set-ups were employed for application; both are shown in Fig. 15. The left part of Fig. 15 shows the bench-2 partition we already used in the previous section, based on gauging network, heights, and land cover. In the right part of Fig. 15 our proposed subdivision of the catchment, based on gauging network and pore volume, is shown. The gauging network has been used as a pre-partition of the basin and the ACS refined the sub-basin density and defined zones for all sub-basins. We chose pore volume as the catchment characteristic. Its spatial organization has been incorporated into the spatial set-up of the model. Our decision was based on the fact that ACS worked better for pore volume than for slope and that information about storage capacities seemed to be more valuable for a conceptual (storage-based) model.

Spatial structures for the HBV

Besides the incorporation into the model structure, we were able to use information about the catchment characteristics in the calibration process. Each (ACS) sub-basin featured three zones, each having an individual average value for pore volume. Since we minimized the heterogeneity of the respective characteristic we were able to assume a uniform distribution of this value for the entire zone. Now, an automated calibration routine benefitted from this information and we were able to reasonably couple parameters of the zones by their respective average of the characteristic.

Say each zone included a parameter called

As already mentioned we performed calibration by the BOBYQA algorithm (Powell, 2009) progressively from the headwaters towards the outlet of the basin. Both benchmark calibrations employ the same spatial set-up with 38 sub-basins and an average of 30 zones per sub-basin. Due to the different parameterization strategies a different but equally high number of parameters are subject to the calibration (see Table 4). The ACS structure employs 44 sub-basins but only three zones (except for two sub-basins having only one zone), giving 51 parameters per sub-basin. Compared to the benchmark calibrations this number is 6 to 9 times lower. A high number of parameters is assumed to offer a model structure with a higher flexibility to match the observed data, though its higher complexity might lower its performance. To compare our proposed model structure with a benchmark at a similar number of parameters, we added a third calibration strategy. The performed approach coupled all zonal parameters as described above. This lowered the number of parameters per sub-basin to 45 for both model set-ups. As can be seen in Table 4, the total number of parameters in the benchmark partition is higher than in the new ACS-based partition.

Parameter quantities.

After the calibration (time period 1995–2006) we evaluated the model's performance in three validation periods, two in direct (temporal) proximity to the calibration period and the last at the very beginning of the time series. Model performance has been calculated as the average Nash–Sutcliffe efficiency (NSE) (Nash and Sutcliffe, 1970) of all gauges and is tabulated in Table 5. Results show that ACS parameterizations are superior in all cases. Its increase in performance ranges from 17 to 52 %, in comparison to the free benchmark, 11–21 % to the six-parameter-coupled benchmark, and 5–19 % to the all-coupled parameterization.

Nash–Sutcliffe efficiency of the ACS-based model and benchmark model; six coupled parameters.

Nash–Sutcliffe efficiencies of the benchmark and ACS model.

Besides this “lumped” evaluation, we compared the performance of the models at each gauge in each period. A comparison of NSE for a six-parameter-coupled model is shown in Fig. 16, and for ACS and a free-benchmark model in Fig. 17. Comparison for the all-coupled parameterization is shown in Fig. 18. We can see that the individual performances led to the same conclusion as the lumped performance, though some results are better for benchmark models (both parameterizations). To be more precise, in the case of the six-parameter-coupled model, 20 points (representing a single gauge in one of the time periods) are below equivalency (representing a better performance of the benchmark model), in the case of the free-benchmark model, 12, and for the all-coupled benchmark, 23 points, representing 15, 9, and 20 % of the evaluated cases.

In conclusion we had to ask ourselves: what is the result of this modelling study? Obviously, we could improve the modelling performance. In accordance with findings in the literature we could prove that additional information relevant to hydrological processes improves the model performance (Finger et al., 2015; Li et al., 2015) and, furthermore, we can confirm that the spatial organization of catchment characteristics (in this case pore volume) is relevant information. The latter conclusion is drawn from the fact that the ACS model offers a similar (or superior) model performance to the coupled model although it included fewer model parameters.

Nash–Sutcliffe efficiency of the ACS-based model and benchmark model; free parameterization.

The intention of this study was to assess the spatial organization of catchments and their characteristics as well as to evaluate the benefit we can gain from this information for use in conceptual, semi-distributed hydrological models. First, we proposed the distance-factor function to assess the interaction of an arbitrary catchment characteristic with the flow path. Graphical representations of this function are capable of visualizing the heterogeneity of the considered characteristic. We further build an algorithm on this proposed function, mainly focusing on factor functions of standard deviations, with the objective of reducing the heterogeneity of the respective catchment characteristic. The proposed ACS algorithm utilizes three different techniques to reduce the heterogeneity that were developed by looking at the main sources of heterogeneity visible in natural catchments. The outcome of the algorithm offers a spatial subdivision of the catchment, at a minimum of standard deviation of the respective characteristic.

After the introduction of these methods we performed an extensive test of the ACS algorithm. First, we tested model functionality and its limits of application to four different basins. We evaluated the spatial patterns we obtained relative to visible spatial patterns in the basins. Furthermore we compared the reduction of variance for different characteristics and basins. Next, we evaluated the usefulness of the obtained results. On the one hand we compared the variance reduction to a benchmark separation and on the other hand we merged our results to a semi-distributed hydrological model. The modelling study demonstrated the benefits we can generate from incorporating the spatial organization of the basin.

Nash–Sutcliffe efficiency of the ACS-based model and benchmark model; all zonal parameters coupled.

We were able to confirm that the distance-factor function is a useful tool for detecting non-random spatial patterns and the interaction of catchment characteristics with the flow path. Furthermore, we could confirm its ability to detect anomalies in the structure of the catchment, e.g. spots of different soil types that do not follow the co-evolutional structure of the basin.

The proposed ACS algorithm provided satisfactory results for different catchment forms, sizes, and patterns. The heterogeneity of characteristics with spatial patterns (like soils) turned out to be beneficial for the application of the ACS algorithm in terms of variance reduction. For more fragmented characteristics (like surface slope) displaying a small-scale but spatially equally distributed heterogeneity, the algorithm will certainly provide a subdivision and zonal classification, but the variance reduction was at a comparable level to common approaches for basin subdivision. In addition we identified the general basin shape as influential for the efficiency of the algorithm. Although this is quite obvious, we learned that basins arranged along a single axis (like a strict south-to-north orientation), with variance of catchment characteristics highly correlating with the distance to the outlet, are more difficult to assess for the proposed algorithm.

Our future work will focus on two topics: on the one hand, we have to further improve the subdivision algorithm. At this point we are able to assess the structure of a single characteristic, while it is highly desirable to consider multiple characteristics. Additionally we will have to develop methods to encompass soil enclosures and fragmented characteristics. The latter problem might lead to the well-known HRU concept. We have to study whether such development is desirable. On the other hand, we will address the value for catchment similarity studies. Following intentions by Mesa and Mifflin (1986), who suggested the width function as an indicator of catchment similarity, it might be worthwhile to investigate how results of the distance-factor function can be used to characterize similarity.

Python code and Toolboxes for common GIS-Software products
of the proposed ACS algorithm are available at

Regions of small variance have no need for further subdivisions; hence, they
are detached from the rest of the basin. Since the exact allocation of these
regions is known, all cells within can be defined as target area T (hatched
in Fig. 7a). The remaining cells are drawn together as non-target area NT. If
one random point of the basin is selected as a possible separation point (SP)
and its watershed is calculated, the set of points belonging to the
watershed, or sub-basin, of SP, BSP, is obtained. The calculated watershed
BSP covers parts of T and NT and hence a coverage rate can be calculated as
the proportion of the cardinalities of the intersections and their respective
superset:

The objective of a detachment O is to find a separation point (SP) whose basin BSP covers a maximum of T and a minimum of NT. Please note that for regions located at the outlet of the basin or at its upstream boundary, only one SP will be defined. Possible SPs are assumed to be allocated at the transit of the main stream from T to NT, or vice versa. An iterative search returns the coverage values O and the highest value is selected as SP, defining a new sub-basin. In the upper part of Fig. 7a the obtained separation as well as the rejected SPs of the iteration (hollow points) are shown.

Distance-factor function of flow accumulation in the catchment of the Mulde.

To identify branches, the distance-factor function of flow accumulation (i.e. the cumulative number of cells draining into a cell) (FAcc) is examined. FAcc indicates the contributing drainage area to each stream cell. Hence, discontinuities in the distance-factor function indicate confluences of streams (see Fig. A2 for an example of the distance-factor function).

Beginning at the outlet (0 on the

Drainage points of major streams or major stream branches are identified
likewise. Before the objective function is called, prevailing FAcc values in
the basin are checked. If the FAcc value of the tributary stream is higher
than the threshold value

The iterative search for the optimal zonal classification involves three
parameters: reduction of Strahler order

After each iteration the average, distance-based standard deviation (Eq. 6)
is calculated. The parameter combination giving the lowest

The authors declare that they have no conflict of interest.

The authors would like to thank the editor and all reviewers for their comments that helped to improve this article. Edited by: Günter Blöschl Reviewed by: three anonymous referees