Implementation and testing of routing algorithms in the distributed Hydrologiska Byråns Vattenbalansavdelning model for mountainous catchments

The main purpose of this study was to implement and test routing algorithms in the distributed Hydrologiska Byråns Vattenbalansavdelning (HBV) model with the emphasis of obtaining a most suitable routing algorithm for large mountainous catchments. Two routing algorithms were built into the grid-based HBV model and tested on the Losna (11,213 km) and the Norsfoss (18,932 km) catchments in central southern Norway. In the first algorithm, runoff is first routed from cell to cell and hydrographs are re-calculated at each cell, and then runoff is routed by the Muskingum–Cunge method in river channels. The second algorithm is a source-to-sink method, which routes runoff of all cells to the catchment outlet as a function of local slope and a calibrated velocity parameter. The routing approaches were compared at different spatial resolutions (i.e. 1, 5 and 10 km) in daily streamflow simulation. Additionally, the elevation band-based semi-distributed model was also compared with the distributed models. The results show that the distributed HBV models are able to perform better than the elevation band-based model, and hillslope routing is crucial in the mountainous catchments. However, incorporating the Muskingum–Cunge channel routing does not add value to the simulation of daily runoff in the mountainous catchments. doi: 10.2166/nh.2013.009 om http://iwaponline.com/hr/article-pdf/45/3/322/372700/322.pdf 2020 Hong Li Chong-Yu Xu Department of Geosciences, University of Oslo, PO Box 1047 Blindern, 0316 Oslo, Norway Stein Beldring Norwegian Water Resources and Energy Directorate, PO Box 5091, Majorstua, 0301 Oslo, Norway Chong-Yu Xu (corresponding author) Department of Earth Sciences, Uppsala University, 75236 Uppsala, Sweden E-mail: chongyu.xu@geo.uio.no


INTRODUCTION
There are two main aims of simulation models, i.e. to explore the implications under certain assumptions about the nature of the real world system, and to predict the behaviour of the real world system under a set of naturally occurring and/or human induced circumstances (Beven ). Hydrological models provide a framework to investigate relations between climate and hydrologic response, and are widely used in stages, from input-output black box models, through lumped conceptual models to physically-based (or conceptual) distributed models (Singh ). Due to more explicit representation of hydrological processes and larger parameter space, distributed models are expected to perform better than lumped models. However, the value of the distributed modelling approach cannot be highly recognized if the sole objective is to simulate discharge at the catchment outlet (Beven ; Wrede et al. ). This statement is partly limited to past research and data availability as well as the designed model In most distributed hydrological models, runoff generation is considered as vertical process, and water flows in one dimension in river channels. The isolation between the spatially distributed pixels is a fundamental limitation to their use in defining today's model structures (McDonnell ). The increasing attention in variable source area runoff generation concepts (Anderson & Burt ; Frankenberger et al. ; McDonnell ) shows that the horizontal processes can also contribute to the runoff generation. It is necessary and worthwhile to develop a fully distributed hydrological model system to include the horizontal processes, which are of importance in runoff routing.
In distributed models, routing is usually dealt with in two ways, i.e. cell-to-cell and source-to-sink. The cell-to-cell approach, like the Muskingum method, consists of determining the amount of water that flows from land cells to its neighbor downstream river cell and tracking the water over the river network. Source-to-sink approach is referring to routing from 'cells in which runoff is produced' to the 'water- Gong et al. ). The main disadvantage is that the performance of cell-to-cell routing is sensitive to the spatial resolution, since this routing method is based on the river network which is extracted from a digital elevation model (DEM) (Gong et al. ). Unlike the cell-to-cell routing, the source-to-sink method originates from the travel time, which can go back to the time-area unit hydrograph theory. The travel time is the quotient of distance and flow velocity. The distance is the length of flow path, which is easily calculated from DEM. Assuming the time-independent flow velocity parameterized as a function of slope, Gong et al. () reported that source-to-sink method gave stable results independent of spatial resolution, while the performance of a linearreservoir-routing method, which is a typical cell-to-cell method, declined as the spatial resolution became coarser.
Additionally, the source-to-sink method was relatively computation effective. Actually, the sensitiveness of the cell-to-cell method is caused by the river network, not the routing method itself. Unlike the source-to-sink methods which only build relationships between all the grids within the catchment and the outlet, the cell-to-cell methods provide opportunities to examine the effects of water movement between the hillslope surfaces by subdividing the hillslope into a series of discrete spatial cells.
In this research, both cell-to-cell routing method and source-to-sink method are incorporated into the Hydrolo-  The main objective of this study is to implement different types of routing methods into a grid-based HBV model and to compare the new model with the semi-distributed HBV model at different spatial resolutions. This study should not only contribute to the overall understanding of the effects of spatial discretization in conceptual rainfall runoff modelling, but also provide an improved modelling tool for flood forecasting and water resources assessment.

Study area
The Glomma catchment, with a basin area of 41,963 km 2 , is located in central southern Norway, and covers around 15% of the area of Norway (Figure 1). With two main branches, the Glomma River is the longest river and has the largest drainage basin area in Norway. Approximately, 30% of the catchment is situated above 1,000 m above mean sea level and 40% between 500 and 1,000 m above mean sea level. Climate varies considerably along the Glomma River from upper glacial regions in the northwest to the lowlands in the south. The northern part is characterized by lower temperature, more precipitation and more snow than the southern part (Tockner Records  from the meteorological station in Lillehammer (226 m above mean sea level) showed the mean annual temperature is 2.9 W C, ranging from À9.3 to 14.7 W C in January and July, respectively. The mean annual precipitation is 720 mm, of which more than  Details of the method will be discussed in the later sections.  Table 1.

Model setups
This study compared six different model setups, including a semi-distributed model, i.e. elevation band-based HBV and a grid-based HBV without routing, and four grid-based distributed HBV models with different routing methods.

Semi-distributed models
Two semi-distributed models include the following. (1)

Distributed models
According to the routing in the hillslope and the river cells, four grid-based distributed models are defined as follows.
(1) DirectM only considers distributed routing in the rivers, and runoff generation is performed in every grid.
The runoff is summed at the respective river cell and is routed by the Muskingum-Cunge method between the river cells. The river channel shape is assumed to be rectangle. When the Courant number is greater than one, the output discharge is equal to input discharge, which means no routing. Details of Muskingum-Cunge method can be seen from, for example, Todini ().
(2) Drain0 only considers the distributed hillslope routing without routing in the river, which is calculated following Equations (1) and (2). First, the runoff generation subroutine runs in the most upslope cells, and the generated runoff is added to the soil moisture storage and subsequently to groundwater  Table 2.
where SM: soil moisture storage; NetP: net precipitation; R: grid total runoff; RU: grid runoff from the upper zone; RL: grid runoff from the lower zone; i, j: grid index; n: the number of the upslope cell of the jth cell.
where S i : ith grid slope in the flow path; V: calibrated water velocity parameter; V i : grid velocity as a function of V and slope length; l i : ith grid slope length in the flow path; t: travel time of one grid.

RESULTS
The six models defined in the previous chapter were calibrated during 1981-1990 and validated during 1991-2010 both for the Losna and Norsfoss stations. The models based on square grid were calibrated and validated at 1, 5 and 10 km horizontal resolutions, respectively. The Nash-Sutcliffe efficiency (NSE) (Nash & Sutcliffe ) and relative mean error (RME) were used as the criteria for model performance. Their formulas are shown in Equations (5) and (6).
where O i and S i are the observed and simulated flows, respectively; i is the time series index, and n is the total number of time steps.
As shown in Table 3, the RME of all the models are less than 3% with an exception in the Losna catchment at the 10 km spatial resolution for the Direct0 and NRF models.
Only the results of NSE are discussed in the following sections since RME is small for all the models, as can be seen in Table 3.
The five grid-based distributed model setups could be grouped to two categories -'Drain' and 'Direct'according to whether or not involving land routing process, which means water interactions between the landscape cells.
Direct0, DirectM and NRF are in the 'Direct' category.

Model comparison
The results of NSE are shown in Figure 3. The following can be seen from the figure. ( or 'Drain' category shows that the channel routing does not add values in the daily flow simulation in the mountainous catchments in Norway.

Spatial resolution effects
Sensitivity of distributed models to spatial resolution is an important characteristic of model robustness. The grid based models were compared in Figure 4 for three spatial resolutions, i.e. 1, 5 and 10 km.
The following can be seen from Coarser resolution leads to low quality input data, which is expected to result in decreasing model performance.

DISCUSSION
One strong advantage of NRF is that the method can tell the travel time of every grid to the catchment outlet, Although the calibrated water velocity parameter of the Losna catchment is smaller than that in the Norsfoss catchment, the Losna catchment has a much faster response than the Norsfoss catchment ( Figure 6). Figure 6 shows that most of the runoff would flow out within 2 days, which is reasonable due to the high slope. Therefore, the time delay caused by routing in the river networks is difficult to present in daily models. However, the flow velocity is only simplified as a function of a calibrated water velocity parameter and slope due to the lack of detailed and high quality data of soil   the horizontal distribution of input data, the water interactions between the landscape cells, and the distributed routing in the river network; NRF considers the horizontal distribution of input data and the time delay between the grid cells and the catchment outlet. In the development of distributed hydrological models, the same runoff generation processes as in lumped models can still be used, and the problem arises as to how to make the distributed elements communicate with each other.
In this research, the hydrological models are only evaluated according to the daily river flow simulation in terms of NSE and RME. In regulated catchments, even well structured hydrological models for natural condition are not easily able to give good simulation. Besides, the good fitness

CONCLUSIONS
In this paper, a grid-based distributed HBV model with two widely used routing methods, a cell-to-cell method with hillslope routing and river routing, and a source-to-sink method were tested in the two sub-catchments of the Glomma basin at 1, 5 and 10 km spatial resolutions. The grid-based distributed HBV models were able to perform remarkably better than the semi-distributed elevation band-based model. The results show that the main discharge response was controlled by the water interactions between landscape cells and taking that into account can improve the daily flow