We present HYPERstream, an innovative streamflow routing scheme based on the width function instantaneous unit hydrograph (WFIUH) theory, which is specifically designed to facilitate coupling with weather forecasting and climate models. The proposed routing scheme preserves geomorphological dispersion of the river network when dealing with horizontal hydrological fluxes, irrespective of the computational grid size inherited from the overlaying climate model providing the meteorological forcing. This is achieved by simulating routing within the river network through suitable transfer functions obtained by applying the WFIUH theory to the desired level of detail. The underlying principle is similar to the block-effective dispersion employed in groundwater hydrology, with the transfer functions used to represent the effect on streamflow of morphological heterogeneity at scales smaller than the computational grid. Transfer functions are constructed for each grid cell with respect to the nodes of the network where streamflow is simulated, by taking advantage of the detailed morphological information contained in the digital elevation model (DEM) of the zone of interest. These characteristics make HYPERstream well suited for multi-scale applications, ranging from catchment up to continental scale, and to investigate extreme events (e.g., floods) that require an accurate description of routing through the river network. The routing scheme enjoys parsimony in the adopted parametrization and computational efficiency, leading to a dramatic reduction of the computational effort with respect to full-gridded models at comparable level of accuracy. HYPERstream is designed with a simple and flexible modular structure that allows for the selection of any rainfall-runoff model to be coupled with the routing scheme and the choice of different hillslope processes to be represented, and it makes the framework particularly suitable to massive parallelization, customization according to the specific user needs and preferences, and continuous development and improvements.

The increasing pressures on freshwater resources originating from a multitude
of complex and interacting factors led in recent years to a growing need of
tools able to provide water resources information at regional to global
scales (see, e.g.,

The hydrological component of both categories of models, which for simplicity
we indicate here as LHMs, relies on simplified conceptualizations and empirical
upscaling procedures

Hyper-resolution LHMs relying on global digital drainage networks at fine
scales, such as HydroSHEDS at 90 m resolution

Mixed schemes in which routing is separated from runoff have also been
employed

To overcome the above limitations without resorting to hyper-resolution
hydrological models (when not needed to better reproduce spatial variability
of soil water storage and transmission), we propose a multi-scale approach
for streamflow routing based on the travel time approach. More specifically,
we propose a scheme based on the width function instantaneous unit hydrograph
(WFIUH) theory

An additional crucial aspect is the computational time efficiency, which stems from the fact that the most demanding step of the procedure is the computation of the geomorphological width functions, which is performed only once as an offline pre-processing procedure. This characteristic, coupled with easiness of parallelization and parsimony in parameterization, inspired us to coin the name HYPERstream, where “HYPER” recalls that the proposed model is based on a “HighlY Parallelizable and scalablE Routing” scheme. Finally, HYPERstream is designed with a flexible and modular structure which allows the coupling with any lumped or process-based formulation for infiltration and subsurface flow processes, while the simplicity and computational efficiency makes it an appealing tool for uncertainty assessment of the predictions, and in general for simulations conducted in a Monte Carlo framework.

This paper is organized as follows: Sect.

As stated in the Introduction, our aim is to develop a simple, parsimonious
and computationally efficient method for streamflow routing (with particular
attention to floods) in large river basins. To this aim, we adopt different
modeling strategies for the river network and the associated hillslopes (the
land component introduced in Sect.

The sketch of Fig.

Sketch of basin conceptualization: subdivision of the study area
into macrocells and nodes (red dots). River network is subdivided into
hillslope–channel transition sites (colored squared symbols) each associated with a pertaining
hillslope area

Drainage characteristics of the basins are obtained from the DEM of the area of
interest. The spatial resolution of the DEM should be fine enough to adequately
capture the spatial structure of the drainage basins and the embedded river networks.
Following procedures widely adopted for the identification of drainage direction and
hillslope–channel separation

The area of the hillslopes changes through the domain, unless identification
of the river networks is performed by using a constant threshold area. The
link between the hillslope and the channel is denoted as hillslope–channel
transition site. As an example, a synthetic DEM grid is shown in Fig.

The next step in the construction of the model is the identification of the

We denote with

Whatever the hillslope model, for the sake of generality hereafter we
indicate with

According to the above conceptual scheme, water flow produced by the hillslope enters the network system through the hillslope–channel transition site and is subsequently routed through it.

From this kinematic scheme, it follows that the streamflow contribution of the
hillslope

Under the hypothesis that the stream velocity

The streamflow

Finally, water discharge

In the first right-hand term of Eq. (

Maps showing

The above method is simple and computationally effective. The underlying
principle is similar to the block-effective dispersion employed in
groundwater hydrology

Routing requires the definition of only a parameter, the channel velocity

In this section we describe an application of HYPERstream to the
Upper Tiber river basin, providing a practical example of model characteristics and performances.
The study area covers the upper
portion of the Tiber river basin, located along the Apennine ridge (central
Italy) between

Intense precipitation events
are typically associated with humid frontal advection from the Mediterranean
Sea and condensation due to the orographic uplift. Because of strong
topographic gradients, headwaters experience intense rainfall events, mostly
occurring from autumn to spring, associated with frequent flood events.
Substantial flood events have been also observed in the floodplain of
the river (southern part) where most of population and economical
activities are clustered

Main geomorphic characteristics of the inter-basin drainage areas within the Upper Tiber river basin (CV: coefficient of variation).

The control sections adopted for multi-site model validation (see Sect.

Map showing the subdivision of the watershed into five inter-basins, each one identified by a node where water discharge is computed (black triangles). The locations of the meteorological stations are also shown as colored dots.

In the following the effect of spatial discretization on the
hydrologic response is analyzed with reference to macrocells of different
dimensions. In particular, the study area was overlaid with macrocells of
increasing size, from 1 to 150 km (the latter including the whole Upper
Tiber river basin within a single macrocell), and corresponding to about 0.009 and 1.25

The identification of the drainage network and associated geomorphic metrics was
performed by adopting standard DEM pre-processing techniques. In particular,
the identification of the flow path lengths involved the following steps:
(i) pit and flat area removal following the procedure of

For a given resolution of the macrocell grid, it is thus possible to derive
the frequency distribution

Figure

Width functions computed at the Santa Lucia (SL) control section for
selected macrocells (colored lines) and for the whole sub-catchment (grey
lines, panels at the bottom), considering grid sizes of

Width functions of the Upper Tiber river basin at Ponte Nuovo (PN)
outlet (4116 km

In this section we present an example of application of HYPERstream for flood
prediction in the Upper Tiber basin, with the purpose to illustrate its major
computational and functional features. To focus on the routing scheme, the
exercise has been intentionally kept as simple as possible. In particular,
the hillslope production function has been defined by combining the widely
used SCS-CN method

At the hillslope scale runoff is computed by using the classic SCS-CN scheme:

Therefore, the effective rainfall intensity

The specific water flux produced by the hillslopes of the macrocell

In order to illustrate model performance we selected two major rainfall
events within the decade 1990–2000, which generated significant, yet not
extreme, floods. The streamflow triggered by these rainfall events was
compared with observational data at the five nodes described in Sect.

Optimal model parameters, calibrated at Ponte Nuovo station (event February 1999), Nash–Sutcliffe efficiency indexes for Ponte Nuovo and all nodes, and computational time cost (for 100 000 runs) resulting from the calibration procedure, for different spatial scale resolutions (size of the macrocell).

In order to test the computational efficiency of HYPERstream, model calibration
was performed generating a large number (i.e., 100 000) of model parameter
sets using the Latin hypercube sampling technique

Comparison between

In all cases, the NSE index at the calibration section (PN) assumes high
values, close to one, indicating a very good model fit to the observed
streamflow data. Optimal parameter sets assume similar values at all the scales, suggesting that the model is able to preserve
geomorphological dispersion when the domain is discretized with macrocells
of increasing dimension. This is verified also when a single macrocell of 150 km resolution is used, though in this case the impossibility to
reproduce the spatial variability of the rainfall (given that only a single
macrocell is used the precipitation is considered uniform over the entire
basin) resulted in an inaccurate description of inter-basin propagation of
fluxes, as emphasized by the negative values of the NSE index averaged over
all nodes. Conversely, for all the other spatial resolutions, overall NSE
values between 0.47 and 0.65 were obtained. Notice that all cases with the
average NSE

Model validation was carried out by means of a combined multi-site,
multi-event approach and was coupled with a Monte Carlo-based uncertainty
analysis performed on a subset of parameter combinations sampled during calibration
at PN with the 1999 flood event as observational data. This subset was identified according to a model
efficiency rejection criterion that classifies as behavioral all sets of
parameters with a NSE index greater than zero, resulting in 21 501 parameters sets
and model
realizations. Successively, the 95 % uncertainty bands associated with the
retained simulations were evaluated using the standard likelihood weighted
procedure proposed by

This work presents an innovative, multi-scale streamflow routing method based on the travel time approach. The principal aim is to develop a simple, parsimonious and computationally efficient method for modeling streamflow (and particularly floods) in large basins. The model, coined as HYPERstream, aims to correctly reproduce the relevant horizontal hydrological fluxes across the scales of interest, from a single catchment to the whole continent. The method is based on the WFIUH theory applied to a hybrid raster–vector data structure, which allows the derivation of localized information on travel times and flow characteristics without the need of narrowing the resolution of the computational grid adopted for the study area. The relevant features of the model are illustrated through the modeling of two flood events in the Upper Tiber river basin (central Italy), with four different domain discretizations, i.e., different dimensions of macrocells.

The main results of the present work can be summarized as follows.

HYPERstream employs a strategy for modeling cell-scale runoff dispersivity such that the simulation of horizontal hydrological fluxes is independent of the grid size, which in turn is a function of the resolution of the atmospheric model or the integral scale of observed precipitation (in case ground-based rainfall measurements are used as in the example application provided here). In particular, the contribution of the geomorphological dispersion is kept invariant at all spatial scales, since in our scheme river network response is derived from the morphological information embedded in the available DEM. This “perfect upscaling” characteristic of HYPERstream is particularly important in all cases when the catchment response needs to be accurately represented, e.g., when dealing with extreme events like floods and inundations.

The above “perfect upscaling” characteristic allows adopting large cells, making the model suitable to large-scale models, up to the continental scale. The overall response function of the river networks will anyway be preserved, no matter the discretization.

Computational efficiency is another relevant feature of the proposed approach. Efficiency stems from the fact that the demanding calculation of the width functions is a pre-processing, one-time effort. Furthermore, the model is prone to parallelization, stemming from the linearity of routing and independency of the runoff generation module adopted at the cell scale. These features make HYPERstream an appealing tool for uncertainty assessment of the predictions, and for simulations conducted in a Monte Carlo framework.

The routing component of the model (including hillslope routing) depends on two parameters, with the additional parameters inherited from the conceptual model of runoff generation adopted at the hillslope scale. While in principle no limitations are posed to the latter conceptualization, we are in favor of a pragmatic “downward” approach, which limits the total number of parameters, to reduce uncertainty and overparameterization. Parsimony is important for a meaningful and reliable parameter estimation procedure and uncertainty analysis.

We believe that all of the above characteristics make HYPERstream an appealing routing tool to be implemented in LHMs, particularly suitable for climate change impact studies where the accuracy of the streamflow routing may be significantly affected by the spatial resolution adopted.

This research has been partially funded by the Italian Ministry of Public Instruction, University and Research, through the project PRIN 2010–2011 (Innovative methods for water resources management under hydro-climatic uncertainty scenarios, prot. 2010JHF437). S. Piccolroaz, B. Majone, and A. Bellin were also supported by European Union FP7 Collaborative Project GLOBAQUA (Managing the effects of multiple stressors on aquatic ecosystems under water scarcity, grant no. 603629-ENV-2013.6.2.1). Authors also thank the Hydrographic Service of Umbria Region for data provision. Edited by: R. Moussa