Journal of Hydrology 368 (2009) 237–250
Contents lists available at ScienceDirect
Journal of Hydrology
journal homepage: www.elsevier.com/locate/jhydrol
Large-scale runoff routing with an aggregated network-response function
L. Gong a,*, E. Widén-Nilsson a, S. Halldin a, C.-Y. Xu a,b
a
b
Department of Earth Sciences, Uppsala University, Villavägen 16, SE-752 36 Uppsala, Sweden
Department of Geosciences, University of Oslo, P.O. Box 1047, Blindern, NO-0316 Oslo, Norway
a r t i c l e
i n f o
Article history:
Received 9 April 2008
Received in revised form 3 December 2008
Accepted 7 February 2009
This manuscript was handled by
K. Georgakakos, Editor-in-Chief, with the
assistance of Ana P. Barros, Associate Editor
Keywords:
Large-scale
Linear reservoir
Routing
Scale independence
Network-response function
Water balance
s u m m a r y
The accuracy of runoff routing for global water-balance models and land-surface schemes is limited by
the low spatial resolution of flow networks. Many such networks have been developed for specific models
at specific spatial resolutions. However, although low-resolution networks can be derived by up-scaling
algorithms from high-resolution datasets, such low-resolution networks are inherently incoherent, and
slight differences in their spatial resolution can cause significant deviations in routing dynamics. By
neglecting convective delay, storage-based routing algorithms produce artificially early arriving peaks
on large scales. A theoretical comparison between a diffusion-wave-routing algorithm and linear-reservoir-routing (LRR) algorithm on a 30-km cell demonstrated that the commonly used LRR method consistently underestimates the travel time through the cells. A new aggregated network-response-function
(NRF) routing algorithm was proposed in this study and evaluated against a conventional flow-net-based
cell-to-cell LRR algorithm. The evaluation was done for the 25,325 km2 Dongjiang (East River) basin, a
tributary to the Pearl River in southern China well equipped with hydrological and meteorological stations. The NRF method transfers high-resolution delay dynamics, instead of networks, to any lower spatial resolution where runoff is generated. It preserves, over all scales, the spatially distributed time-delay
information in the 1-km HYDRO1k flow network in the form of simple cell-response functions for any
low-resolution grid. The NRF routing was shown to be scale independent for latitude–longitude resolutions ranging from 50 to 1°. This scale independency allowed a study of input heterogeneity on modelled
discharge modelled with a daily version of the WASMOD-M water-balance model. The model efficiency of
WASMOD-M-generated daily discharge at the Boluo gauging station in the Dongjiang basin in south
China was constantly high (0.89) within the whole range of resolutions when routed by the NRF algorithm. The performance dropped sharply for decreasing resolution when runoff was routed with the
LRR method. The three WASMOD-M parameters were scale independent in combination with NRF, but
not with LRR, and the same parameter values gave equally good results at all spatial resolutions. The
effect of spatial resolution on the routing delay was much more important than the spatial variability
of the climate-input field for scales ranging from 50 to 1°. The extra information in a distributed versus
a uniform climate input could only be used when the NRF method was used to route the runoff. NRF
requires more labour than LRR to set up but the model performance is very much higher than the LRR’s
once this is done. The NRF method, therefore, provides a significant potential to improve global-scale discharge predictions.
Ó 2009 Elsevier B.V. All rights reserved.
Introduction
Large-scale routing algorithms transfer runoff to discharge in
global and continental water-balance (Vörösmarty et al., 1989;
Döll et al., 2003) and land-surface models (Russell and Miller,
1990; Liston et al., 1994; Coe, 2000; Arora, 2001; Hagemann and
Dumenil, 1998). Runoff routing at large scales normally involves
development of low-resolution flow networks, the spatial resolutions of which range from 1 km (HYDRO1k, USGS, 1996a) to
* Corresponding author. Tel.: +46 18 471 2521.
E-mail address: lebing.gong@hyd.uu.se (L. Gong).
0022-1694/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.jhydrol.2009.02.007
4° 5° latitude–longitude (Miller et al., 1994). Many global
water-balance models use a 0.5° 0.5° latitude–longitude grid
(Fekete et al., 2002; Arnell, 2003; Döll et al., 2003) since this has
been found suitable for a broad range of global water-resources
and water-quality studies (Vörösmarty et al., 2000a,b). There exist
at least five global routing networks with this resolution (Hagemann and Dumenil, 1998; Graham et al., 1999; Renssen and
Knoop, 2000; Vörösmarty et al., 2000a,b; Döll and Lehner, 2002).
The lack of a common network complicates inter-comparison of
global models, which is regrettable because of the large l differences in model predictions that are seen even when runoff
predictions are aggregated to global and continental scales
(Widén-Nilsson et al., 2007; Hanasaki et al., 2008).
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L. Gong et al. / Journal of Hydrology 368 (2009) 237–250
Most large-scale routing models apply storage-based routing
algorithms on low-resolution flow networks. Such algorithms are
based on mass conservation and relationships between river-channel storage, and river inflows and outflows. In the Muskingum
method (McCarthy, 1939), the storage S is a function of both inflow
I and outflow O:
S ¼ K ½x I þ ð1 xÞ O
ð1Þ
The mean residence time K can be approximated by the time
needed by the wave to travel through the reach, whereas x is a
shape parameter controlling the relative importance of inflow on
the outflow hydrograph. For most rivers, x takes values in the range
between 0 and 0.3 with average around 0.2 (Linsley et al., 1982).
The parameters of the Muskingum equation can be estimated
graphically from inflow and outflow hydrographs or, as shown by
Cunge (1969), from flow hydraulics. Since estimation of x requires
local knowledge for each river reach, it is always set to zero in global-scale applications. This zero x simplification implies that
wedge storage in the channel is unimportant (e.g. Linsley et al.,
1982) and that there are no wave-velocity delays. With this simplification, the Muskingum method reduces to the linear-reservoirrouting method (LRR), used widely on large-scale networks because
of its simplicity. Examples are the routing models by Sausen et al.
(1994), Miller et al. (1994), Liston et al. (1994), the HD model (Hagemann and Dumenil, 1998), TRIP (Oki et al., 1999) and its applications (Oki et al., 2001; Falloon et al., 2007; Decharme and
Douville, 2007), HYDRA (Coe, 2000) and its application (Li et al.,
2005), WTM (Vörösmarty et al., 1989) and its application (Fekete
et al., 2006), RTM (Branstetter and Erickson, 2003), and the routing
model of WGHM (Döll et al., 2003). Although all are based on similar principles, they are named differently, e.g., linear routing, linear
Muskingum routing (Arora and Boer, 1999), and simple advection
algorithm (Falloon et al., 2007). The number of linear reservoirs
sometimes exceeds unity, e.g., two (Arnell, 1999, within-cell routing) or more (Hagemann and Dumenil, 1998). Wave velocity is an
important parameter in most LRR algorithms and is normally obtained by calibration to a downstream discharge time series. The
velocity can be fixed globally (Coe, 1998; Döll et al., 2003) but model performance is improved if it is varied between basins (Miller
et al., 1994; Arora et al., 1999; Döll et al., 2003). Vörösmarty and
Moore (1991) assign a transfer coefficient to each cell on the basis
of geometric considerations, implying a constant wave velocity that
should be calibrated. Fekete et al. (2006) use a temporally uniform
but spatially varying velocity field derived from an empirical relation between mean annual discharge, slope, and flow characteristics
after Bjerklie et al. (2003). Spatially variable velocities were first
introduced by Arora et al. (1999), and further developed by Arora
and Boer (1999) to allow for temporally variable velocity. Hydraulic
equations can be used to relate modelled wave velocities to riverchannel geometry. Such velocities require real river segments that
can be derived from large-scale flow-net segments after multiplication with a meandering factor (Arora and Boer, 1999; Lucas-Picher
et al., 2003).
The main hypothesis in this paper is that storage-based routing
algorithms are not suitable for large-scale river routing. This
hypothesis is based on two arguments. The first is that storagebased routing methods, even sophisticated ones like the Muskingum-Cunge method, lack convective time delay (Beven and Wood,
1993) such that an up-stream input will have an immediate effect
on the downstream output. The convective delay increases with
the length of the reach, so ignoring it may not work well at scales
where both network segments and river lengths are very long. The
second argument is that storage-based routing algorithms are
inherently scale dependent since they rely on flow networks that
change with spatial resolution. On one hand, lower resolution leads
to a decrease of derived slope resulting in longer travel times and
lower peak flows; on the other hand, lower resolution also leads to
a decrease of flow path resulting in shorter travel times and high
peak flows, and these two effects may compensate each other to
some extent. Another effect of lower DEM resolution is the change
in optimal channel threshold values, pertaining relative to the
channel length. In short, a low-resolution network smoothes the
spatial-delay pattern on large scales. Together with neglecting
the convective delay this may decrease routing accuracy at large
scales. Du et al. (2009) studied the effect of grid size on the simulation of a small catchment (259 km2) in the humid region in China
and showed that changes in spatial resolution of the model will
lead to different values of the GIS-derived slope, flow direction,
and spatial distribution of the flow paths, which in turn affect
the model simulation. In their study, three types of DEMs with grid
sizes of 100 m, 200 m, and 300 m were used to simulate the storm
runoffs. They concluded that when grid size is larger than 200 m
the results are poor. Arora et al. (2001) compare runoff routing at
350-km and 25-km scales with the same runoff input and conclude
that discharge is biased at large scales and also more error-prone at
high and low flows. The method by Guo et al. (2004) to scale up
contributing area and flow directions was designed to improve
decreasing model performance with decreasing spatial resolution.
Although the overall performance improves and reaches a maximum at 7.50 , the decreasing trend in model performance remains
for lower resolution.
For a valid routing algorithm, the wave crest should not travel
through a cell within one routing time step. This gives a practical
disadvantage to storage-based routing methods at large scales
since they require a time step much shorter than the time step of
the runoff-generation model (Coe, 1998; Liston et al., 1994; Sushama et al., 2004; Kaspar, 2004). This requirement means that computational demand may be too great when global water-balance
models are built on a finer spatial grid than commonly used today.
Storage-based routing algorithms are computationally expensive
also because they must route the discharge cell-to-cell. However,
when the reproduction of discharge dynamics at a basin outlet is
an important objective, cell-to-cell methods can be replaced by
source-to-sink methods (Naden et al., 1999; Olivera et al., 2000)
that only simulate delays at pre-defined cells, normally co-registered to a runoff gauge or a river mouth. Such methods enable
more efficient computation, which allows the use of higher-resolution flow networks (Olivera et al., 2000) and more sophisticated
routing methods. This computational efficiency allows Naden
et al. (1999) to use the convective–diffusive approximation of the
Saint Venant equations (Beven and Wood, 1993) at the continental
scale.
Even if existing large-scale routing methods produce biased or
erroneous results, and are computationally expensive, these problems are not seen as sufficiently important as long as the lateral
water transport is considered less important than the vertical
land–surface exchange that dominates runoff generation in many
global-scale models (e.g., Olivera et al., 2000). This may, however,
be too simple a view as long as global models can compensate
for a poor spatial representation of routing time delays by calibration in other parts of the system. Yildiz and Barros (2005) found a
strong dependency in the Monongahela River basin between simulated physics and the flow-network resolution, in particular when
a 5-km resolution was used instead of a 1-km resolution. Less subsurface and more surface runoff was simulated as a result of the
lower hydraulic gradients. The runoff-generation mechanism was
inconsistent with observations at the lower spatial resolution,
and not only resulted in a bad fit to observed discharge, but also
in values of the hydraulic-conductivity parameter that were forced
to compensate the lower gradients at this resolution (Yildiz and
Barros, 2005). A scale-independent routing algorithm would alleviate this problem but scale independency of routing in a multi-scale
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L. Gong et al. / Journal of Hydrology 368 (2009) 237–250
framework has rarely been discussed. This study had two objectives: (i) testing our hypothesis of inaccuracy of storage-based
routing at global and continental scales, and (ii) developing a
new simple, scale-independent, accurate, and computationally efficient large-scale routing algorithm.
In the following sections, after a brief introduction of the study
area and data sources, development of a multi-scale hydrological
grid is presented as the basis for application of storage-based routing which is then applied to flow networks at each resolution. The
generation of daily runoff at different resolutions is used as input
to the routing algorithms. The need for a new routing algorithm
is then motivated in a theoretical study of the inaccuracy and
scale-dependency of storage-based routing. Development and
evaluation of the new routing algorithm is then presented, followed by a discussion of possible future development.
Basin, hydrography and runoff generation
River basin, routing network and climate data
Large-scale routing algorithms have primarily been evaluated
on the largest river basins in the world. Such evaluations are subject to problems because river-flow dynamics may be influenced
by insufficient climatic and hydrological data, regulations by dams
and reservoirs, and water abstraction. We used the well-documented Dongjiang medium-size basin in this study to ascertain
that the routing-algorithm properties would not be too disturbed
by unknown influences. The Dongjiang (East River) basin (Fig. 1)
is a tributary of the Pearl River in southern China. Its 25,325 km2
drainage area above the Boluo gauging station is large enough to
retain generality of the result in a study of global hydrology. The
basin has a dense network of meteorological and hydrological
gauging stations and its hydrology is well studied (e.g., Jiang
et al., 2007; Chen et al., 2006, 2007). The climate is sub-tropical
with an average annual temperature of around 21 °C and only
occasional sub-zero winter temperatures in the mountains. The
1960–1988 average annual precipitation is 1747 mm, and the
average annual runoff is 935 mm or 54% of the average annual precipitation. About 80% of the annual rainfall and runoff occur during
the wet season from April to September. The basin presents a complex mixture of Pre-Cambrian, Silurian, and Quaternary geological
formations showing as granites, sandstone, shale, limestone, and
alluvium. The landscape is characterised by 83% mountains and
hills, 13% plains and 3.8% inland water area. The basin is forestcovered at higher altitudes whereas intensive cultivation dominates hills and plains.
We used HYDRO1k (USGS, 1996a), the gridded global network
with the highest resolution publicly available today, to delineate
the basin hydrography. HYDRO1k is derived from the GTOPO30
30” global-elevation dataset (USGS, 1996b) and has a spatial resolution of 1 km. HYDRO1k is hydrographically corrected such that
local depressions are removed and basin boundaries are consistent
Meteorological
25
Log(fa)
10
Precipitation
9
Discharge
24.5
Latitude (degree north)
8
7
24
6
5
23.5
4
3
23
Boluo
2
1
0
113.5
114
114.5
115
115.5
116
Longitude (degree east)
Fig. 1. The Dongjiang (East River) basin, a tributary to the Pearl River in southern China and the locations of the hydro-meteorological stations used in the study. The flow
network is shown by mapping HYDRO1k 1-km pixels with their logarithmic up-stream flow-accumulation area (fa, in km2).
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L. Gong et al. / Journal of Hydrology 368 (2009) 237–250
with topographic maps. HYDRO1k includes numerous hydrologyrelated data layers, such as aspect, flow direction, drainage area,
elevation gradient, compound topographic index, basin and subbasin boundaries, and DEM-derived stream lines. The HYDRO1k
dataset was developed on a Lambert azimuthal equal-area projection in order to maintain uniform grid-cell area.
Daily hydro-meteorological data were obtained for 1960–1988.
The National Climate Centre of the China Meteorological Administration provided data on air temperature, sunshine duration, relative humidity, and wind speed from seven weather stations.
Precipitation data from 51 gauges and discharge data from 15
gauging stations were retrieved from the Hydrological Yearbooks
of China issued by the Ministry of Water Resources (Fig. 1). Potential evaporation was calculated from air temperature, sunshine
duration, relative humidity, and wind speed with the Penman–
Monteith equation in the form recommended by FAO (Allen
et al., 1998).
The multi-scale hydrological grid
We used HYDRO1k to construct a series of grids with gradually
lower spatial resolution to study the scale dependency of routing.
Grid-cell sizes were varied from 50 to 600 in steps of 10 to form 56
different basin network representations. The downstream Boluo
discharge station (Figs. 1 and 2) was registered in the HYDRO1k
flow-accumulation layer by matching coordinates and up-stream
area. The Dongjiang basin was delineated in this way by 25,886
1-km pixels from the HYDRO1k flow-path layer (Fig. 2). This was
taken as the ‘‘true” basin area. The first step to scale up the HYDRO1k basin grid was to delineate all pixels that were needed to
cover all latitude–longitude cells at the 56 spatial resolutions. In
the following, a ‘‘pixel” refers to a 1-km HYDRO1k unit and a ‘‘cell”
refers to a lower-resolution latitude–longitude square that constitutes the large-scale catchment grid. As long as a latitude–longitude (e.g., 300 ) cell had the centre of a HYDRO1k pixel in it, that
pixel was included as representing the basin (at the 300 resolution).
The area of each cell was obtained by counting the number of HYDRO1k pixels in it to avoid areal errors for boundary cells. The basin delineation changed considerably from the highest to the
lowest resolution (Fig. 2) but the basin area remained constant because of the way boundary-cell areas were handled.
Fekete et al. (2001) present the simple and robust flow-network-scaling algorithm (NSA) to rescale fine-resolution networks
to coarser resolutions. NSA provides a consistent way to generate
lower-resolution routing networks from HYDRO1k and it has been
successfully applied to all continents covered by HYDRO1k. We
used NSA to derive a flow network for each spatial resolution.
Precipitation, temperature and potential evaporation were first
interpolated to the original HYDRO1k 1-km resolution. Precipitation was interpolated from the 51 local stations by kriging with linear variogram whereas inverse-distance weighting was used to
interpolate temperature and potential evaporation from the seven
regional stations. This climate input was then aggregated by averaging the interpolated 1-km data to each cell in the respective 56
lower spatial resolutions.
Runoff generation
This study was initiated to supply WASMOD-M (Widén-Nilsson
et al., 2007), one of the simplest global water-balance models, with
a reliable routing algorithm. WASMOD-M is based on the WASMOD (Xu, 2002) monthly conceptual water-balance model, which
has been successfully applied in many parts of the world. Other
global water-balance models route their runoff with a finer timestep than used for runoff generation (e.g., Döll et al., 2003). A newly
developed, daily version of WASMOD-M was used this study. The
daily WASMOD-M used daily climatic input and generated runoff
in each cell with a daily time step. Compared to the monthly WASMOD-M (Widén-Nilsson et al., 2007), the daily version required its
fast and slow-runoff formulations to be modified, in this case to a
non-linear exponential form:
SP ¼ 1 ec1 LM
ð2Þ
F ¼ Pn SP
ð3Þ
S ¼ LM ð1 ec2 LM Þ
ð4Þ
All other parts of WASMOD-M took the same form as in WidénNilsson et al. (2007), e.g. actual evaporation (E):
AW=Ep
E ¼ minfEp ð1 a4
Þ; AWg
ð5Þ
SP is the percentage of each cell area that is saturated, LM is land
moisture (water available in each cell for actual evaporation and
runoff), F is fast runoff, S is slow runoff or base flow, Pn is net rainfall, AW is water available for actual evaporation, Ep is potential
evaporation. c1 [mm1], c2 [mm1], and a4 [] are parameters, all
of which has a potential range from 0 to 1. The equations for Pn,
Fig. 2. The flow net of the Dongjiang (East river) basin at three different spatial resolutions with the down-most discharge gauging station in Boluo (marked as a filled circle)
taken as the reference point. Left: the original 1-km pixel size of the HYDRO1k database. Middle: Network-scaling-algorithm (NSA)-derived flow net derived at 50 resolution.
Right: NSA-derived net at 0.5° resolution. Note the location of the flow-net exit point from the Boluo station in the two aggregated cases.
L. Gong et al. / Journal of Hydrology 368 (2009) 237–250
AW and LM are given by Widén-Nilsson et al. (2007). The daily
WASMOD-M version was used to simulate runoff generation for
each cell in all resolutions.
In a snow-free region like the Dongjiang basin, WASMOD-M had
only three parameters: the evaporation parameter a4, the fast-runoff parameter c1, and the slow-runoff parameter c2. 250 parametervalue sets were obtained by Latin-Hypercube sampling (McKay
et al., 1979) with prior uniform distribution. Initial parameter
ranges were bounded by their potential values. WASMOD-M was
run with the 250 parameter-value sets at each scale and the resulting 250 runoff-generation time series were saved for each scale as
input to the routing models. Computational limitations prevented
us from using more than 250 sets.
Scale dependency of routing dynamics
The LRR method is based on the following assumptions: (i) the
water surface should be level throughout a cell, so that storage is
only a function of outflow; (ii) the slope of the storage–outflow
curve (dS/dQ) should be constant and equal to the mean residence
time of the store, which can be estimated from the time to travel
through the cell. The LRR method is analogous to a bucket with a
hole in the bottom, which produces an exponentially declining
outflow for an instantaneous input. Cell sizes in many globalscale models are 0.5°, i.e., 56 km at the equator, or larger. The
time needed for the water to pass this distance is not likely negligible, i.e., it takes time for water entering a cell to reach its outlet and the contribution to the outlet discharge is not likely to be
instantaneous. The assumption of exponential decay without convective delay may, therefore, be flawed. Under this assumption
the time expected for the simulated wave crest to travel through
the cell should be much less than the residence time. As cell sizes
increase, the convective dispersion requires more time, and the
assumptions behind LRR are less and less fulfilled. Less simplified
storage-based methods like Muskingum-Cunge (Cunge, 1969)
have the same problem of instantaneous influence on the downstream discharge (Beven and Wood, 1993). It is not difficult to account for correct wave-velocity delays by combining linear
reservoir with a linear channel delay component as introduced
by Dooge (1959) and applied in the form of a network width
function by Surkan (1969); however this idea is seldom used in
global water-balance models.
We tested the possible effects of this instantaneous reaction on
LRR routing dynamics on a 30-km cell. Water was released as a
constant-input pulse of either 1-h or 24-h duration at the inflow
side of the cell. Routing through the cell was done both by LRR
and the diffusion-wave approximation of the full Saint Venant
equations. We tested the influence of cell size on the LRR method
by dividing the 30-km reach into 1–10 equidistant sub-sections.
The diffusion-wave approximation assumes that water movement
is dominated by friction, bed slope and a pressure-slope term. The
diffusion-wave approximation simulates attenuation and dispersion explicitly unlike the kinematic-wave approximation that
models attenuation of the flood peak by the dispersive error of a
numeric approximation (e.g., in the Muskingum form). The relevant equation for flow is the convective–dispersive equation:
2
@Q
@ Q
@Q
¼D 2 c
@t
@x
@x
ð6Þ
Up-stream and downstream boundary conditions, as well as an inflow time series must be specified for Eq. (6). The method was applied on the 1-km HYDRO1k network, where the length of a 1-km
reach is small compared to the length of the river channel in the
cell. An analytical solution can be derived for the case of an instantaneous input at the up-stream end of a reach, given the following
initial and boundary conditions:
Qð0; tÞ ¼ I0
241
ð7Þ
Qðx; 0Þ ¼ 0;
for x > 0
ð8Þ
where Q is the flow rate per unit width of the channel (m2/s), D is
the dispersion coefficient (m2/s), c is the wave velocity (m/s), t is
time (s), x is distance (m), and I0 is the inflow rate per unit width
of the channel (m2/s).The analytical solution Qc of Eq. (6) under
the above boundary condition is given by:
Q c ðx; tÞ ¼
c x
I0
xct
xþct
erfc pffiffiffiffiffiffiffiffi
erfc pffiffiffiffiffiffiffiffi þ exp
D
2
4Dt
4Dt
ð9Þ
where erfc is a complementary error function.
Since we tested two cases with constant inflow during 1 and
24 h, the resulting discharge at the output was obtained by subtracting solutions with zero and 1-h or 24-h offsets:
Q 1 ðx; tÞ ¼ Q c ðx; tÞ Q c ½x; ðt 1Þ
ð10Þ
Q 24 ðx; tÞ ¼ Q c ðx; tÞ Q c ½x; ðt 24Þ
ð11Þ
Q(x, t) became a unit response function since we postulated a unitvolume inflow. In the following we call Q1 a 1-h response function
and Q24 a 24-h response function. Ten different D values (1–10
km2/h) were selected such that the resultant hydrograph attenuations were close to those of the LRR method. Wave velocity was
specified to 0.4 m/s (1.44 km/h).
The aggregated network-response function
Two ideas were central when developing the new routing algorithm. The first was to achieve scale-independent routing by upscaling dynamics from the best available resolution rather than
relying on the apparent river-flow network at any coarser resolution. The second was to achieve high computational efficiency
when routing discharge in a large-scale water-balance model.
These ideas were implemented as a three-stage algorithm. Routing
dynamics should first be parameterised in the form of linear response functions. Because these functions are linear they can then
be aggregated to the desired lower spatial resolution. Parameterisation and aggregation should only be done once for each basin
and resolution. The final routing can then be used for any given
time period and runoff model.
The starting point for the algorithm was to register a downstream station to a reference HYDRO1k pixel. A 24-h response
function could then be calculated with the diffusion-wave solution
(Eqs. (9)–(11)) for all up-stream HYDRO1k pixels, and a daily response function could be obtained by integrating the 24-h response function (as demonstrated in the first part of the
‘‘Results” Section). The method could equally well be elaborated
for time steps other than a day. This daily response function was
specific for each pixel, so it was named the pixel-response function
(PRF) and was used as the source of aggregation:
PRFðtd Þ ¼
Z
Q 24 ðtÞ dt
ð12Þ
td
where t is time after runoff generation, td is days after runoff generation, Q24 is the 24-h response function. Each PRF consists of a number of percentages of runoff (p1, p2, . . ., pn) arriving on days
(d1, d2, . . ., dn) following the runoff-generating day. Like other
large-scale water-balance studies, we did not carry out our simulations at the pixel scale. A cell-response function (CRF) was derived
by aggregating all PRFs within a cell and normalising to unit
volume:
CRFðt d Þ ¼
n
1X
PRF i ðt d Þ
n i¼1
ð13Þ
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L. Gong et al. / Journal of Hydrology 368 (2009) 237–250
where n is number of pixels inside the cell. The aggregation from
PRF to CRF transfers distributed delay information at 1-km scale,
in the form of daily network response function, to any lower resolutions as defined by the size of the cell. This is analogous to the
spatial integration of time delay by Beven and Kirkby (1979) to form
a time-delay histogram for an entire sub-basin. Eq. (13) takes the
simplest form of the aggregation by assuming that runoff generation is constant throughout the cell, a condition that could be relaxed to dynamically weight the aggregation by the sub-cell
variation of runoff input. A network-response function (NRF) to a
distributed daily runoff input was then calculated as:
NRFðt d Þ ¼
m
X
Q j CRF j ðt d Þ
ð14Þ
j¼1
where m is the number of cells and Qj is the runoff-generation volume of the jth cell. If runoff input is spatially uniform, the NRF is
analogous to the network width function (e.g., Surkan, 1969), which
is obtained by counting the number of channel reaches at a given
distance away from the outlet (e.g., Kirkby, 1993). The application
of a width function requires detailed river-network data (e.g., Naden et al., 1999) that are not always available on the global scale.
Because stream length decreases with coarser-resolution network,
width functions obtained from global flow networks are systematically shorter than those derived from high-resolution network,
although the bias can be adjusted with a length correction (Fekete
et al., 2001). Eqs. (13) and (14) indicate that although channel response is represented at cell level it still contains all delay information from pixel level. This is equivalent to directly using
contribution area instead of the number of channel reaches at a given distance away from the outlet. The downstream hydrograph is
then obtained by the convolution of NRF with the input runoff time
series. The efficiency of the algorithm comes from the fact that PRFs
and CRFs are all pre-fixed before convolution.
The method, as given above, requires both the dispersion coefficient D and the wave velocity c to be calibrated against downstream discharge observations. To simplify the method, we
assumed that the daily pixel-response function could be obtained
by integrating the 24-h response function with near-zero D. When
near-zero dispersion is assumed, the 24-h response function can be
well approximated by a pure translation of the daily inflow according to the wave velocity c. Consequently, the pixel-response function is reduced to two percentages of arriving discharge (p1, p2) for
two consecutive days (d1, d2), which are sufficiently described by p1
and d1.This simplification considerably lowered the computational
demand while still providing the correct travel time for the discharge. The 24-h response function with near-zero dispersion is a
pure translation of the up-stream inflow with a time shift equal
to the mean residence time, but the pixel-response function offers
flood-peak attenuation by redistributing a 1-day up-stream discharge over 2 (for a pixel) or more (for a cell) days downstream.
The percentage on each day can then easily be calculated if we further assume that runoff generation has no diurnal variation. The
first parameter, the first day of arriving discharge (d1) is the first
of the two consecutive days, after runoff was generated, when
water arrives at the reference pixel (d1 = 1 means that some of
the generated runoff arrives at the reference pixel on the same
day). The second parameter, the first-day percentage of arriving
discharge p1, gives the fraction of runoff reaching the reference pixel in the first of these 2 days. If, for example, the wave-velocity delay is 3000 min, the runoff generated on day 1 will reach the
reference pixel on days 3 and 4. A part p1 = (1440 3 3000)/
1440 = 92% will arrive on day d1 (=3) and the remaining amount,
1 p1 = 8% on day d1 + 1 (=4). The assumption of constant within-day runoff is reasonable on the global scale where within-day
information normally is absent. The assumption has a smoothing
effect on the routed runoff compared to a situation where the
within-day distribution of runoff generation can be inferred, e.g.,
from within-day precipitation data.
The d1 and p1 parameters were directly obtained from the travel
time t, which is a function of the network wave velocity and network topology. In this study, we assumed that the wave velocity
was only a function of slope. Following the delay calculation developed by Beven and Kirkby (1979) for overland-flow routing, we denoted the distances of the flow-path segments from any HYDRO1k
pixel down to the reference pixel as l1, l2, . . ., ln, and the corresponding slopes as tan(b1), tan(b2), . . ., tan(bn). We also postulated a normalised 1-km-network wave velocity (V45) for a slope of
tan(45°). Beven and Kirkby (1979) calculate overland-flow velocity
for a flow segment in Topmodel as:
V i ¼ V 45 tanðbi Þ
ð15Þ
We found velocity to be less sensitive to slope for large-scale water
transport and modified this equation to:
V i ¼ V 45 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
tanðbi Þ
ð16Þ
A similar approach is used by Ducharne et al. (2003). This time-constant velocity led to the following equation for delay time from any
given pixel to the reference pixel:
t¼
n
X
i¼1
V 45 l
piffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
tanðbi Þ
ð17Þ
V45 should be calibrated against observed discharge. There is no
need to use meandering factors for the flow-path-segment distances since V45 represents an effective network speed.
Water velocity normally varies with magnitude of discharge but
a study at River Severn catchment (8.7 km2) shows that non-linear
water velocity can lead to a constant wave velocity (Beven, 1979).
It was not evident that such a constant wave velocity could be
extrapolated to larger scales, but for simplicity the parameter V45
was set to be constant in this study.
The aggregation from PRF to CRF can be demonstrated as follows: consider a case where the low-resolution cell consists of four
1-km pixels (i = 1 4; Fig. 3a) and where runoff generated in each
of the 4 pixels reaches the reference point in 2 consecutive days
with a first-day percentage pi for each pixel. Each pixel has different d and p values. Each of the 1-km pixels covers 1=4 of the low-resolution cell for which the runoff generation is constant. The runoff
is thus assumed to be the same for each of the four 1-km pixels at
each time step, and the discharge percentage for each pixel given
by the sum of first- and second-day percentages must be divided
by 4, the number of 1-km pixels in the cell. The resulting time-delay distribution for the low-resolution cell is given by a 5-day histogram (Fig. 3b).
The CRF derived in this way for a low-resolution cell preserves
the full delay information from all HYDRO1k pixels. It tells explicitly in which days (P 2) the generated runoff reaches the reference
pixel and the percentage of discharge arriving on each day. A CRF
was constructed for each cell at each resolution before routing
started (Fig. 4). When weighted by a spatially distributed input
(Eq. (14)), the CRF was then used to transform the runoff time series to a discharge time series for each low-resolution cell.
Evaluation
The performance of the new routing algorithm was evaluated
against the LRR and a benchmark NRF algorithm as follows. First,
a ‘‘traditional” linear-reservoir-routing (LRR) algorithm and a
benchmark NRF algorithm were derived at the same spatial resolution. The benchmark algorithm calculated delay (t) as a function of
cell-to-cell distance and a velocity parameter (VBEN). The delay t
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L. Gong et al. / Journal of Hydrology 368 (2009) 237–250
a
c*60%
c*40%
Pixel index
4
c*30%
c*50%
c*50%
2
c*20%
c*80%
1
C = 1/(number of pixels) = 1/4
Percentage of arrived discharge
0
b
c*70%
3
1
2
1
2
3
4
5
6
3
4
5
6
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
Delay in days
Fig. 3. Schematic description of a new routing-aggregation algorithm for a low-resolution cell made up of four source-data pixels (index = 1 4). Day 1 = day of runoff
generation. All pixels are routed differently to the reference point. This figure demonstrates a special case that can be explained as: (a) Runoff from pixel one is the quickest
and reaches the reference point by 20% already on the day of generation, whereas runoff from pixel 3 takes the longest time and the first 30% does not reach the reference
point until day four. (b) Runoff from the low-resolution cell reaching the reference point is made up of the weighted (1=4 ) sum of all pixels for each of the 5 days. The resulting
cell-response function gives the temporal distribution of runoff from the low-resolution cell reaching the reference point.
Fig. 4. (a) Distribution of delay times for the Dongjiang (East river) basin derived by Eq. (17) with V45 = 8 m/s, i.e., the times taken for runoff generated in a given 1-km pixel to
reach the reference point, in this case the down-most discharge station in Boluo (marked by an open circle). Aggregated cell-response functions at 50 resolution (b), and at 0.5°
resolution (c). The x-axes (scale 0–8 days) in the cell-response function (bar graphs in each cell) shown in (b) and (c) represent delay and the y-axes (scale 0–1) first-day
fraction of arrived discharge.
was transferred to the d and p parameters in the same way as in
the NRF routing and the result was a low-resolution 2-day NRF
for each cell. Then, all three algorithms were run both with spatially uniform and with distributed climate-input data, and used
similar calibration procedures to identify the best velocities. The
best model efficiency (Nash and Sutcliffe, 1970) for discharge at
the Boluo gauging station was chosen as a performance indicator
across spatial resolutions and methods. The run with uniform climate was done to guarantee that only scale-dependent routing effects influenced the result. The benchmark algorithm was
constructed to guarantee that a comparison between the new
and the ‘‘traditional” algorithms could be attributed to the algorithm differences, rather than to their different network
resolutions.
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L. Gong et al. / Journal of Hydrology 368 (2009) 237–250
Linear-reservoir routing
The NSA-derived flow net was used to route runoff with a LRR
method between cells for all basin grids at 50 –600 . A linear reservoir
model was used at each resolution:
dS
¼ Q in þ Q r Q out
dt
SðtÞ ¼ K Q out ðtÞ
K
dQ out
¼ Q in þ Q r Q out
dt
ð18Þ
ð19Þ
ð20Þ
S is cell storage, Qin inflow from up-stream cells, Qout outflow to a
downstream cell, Qr runoff generation of the cell, and K travel time
from an up-stream cell to a downstream cell. K was estimated, as
commonly done, from the ratio of the cell-to-cell distance and a
wave-velocity parameter (VLRR). The outflow discharge Qout was calculated with a finite-difference approximation (Arora et al., 1999).
imation (Fig. 5c, d). This time difference would show up in
all cells and gradually accumulate. It could, thus, become
substantial for large basins.
(2) The LRR delay pattern did not change monotonically with
cell size, but instead reached a maximum for a cell size of
around 4–5 sub-sections of the total 30-km reach, i.e., for a
reach around 7 km. This could be an indication that such a
cell size could be optimal for the LRR method, although it
did not perform very well even at this spatial resolution
(Fig. 5c).
(3) The daily response function for the diffusion-wave approximation monotonically approached the zero-dispersion solution for small D values. The zero-dispersion response
function was a pure translation of the up-stream inflow with
a time shift equalled to the mean residence time. When this
translation function was integrated to a daily time step, all
discharge arrived in two consecutive days (Fig. 5d).
Calibration of model parameters
Flow velocities and meander factors
Wave-velocity parameters for both NRF and LRR routing were
obtained separately before the calibration of runoff-generation
parameters. At each grid resolution, 250 runoff time series were
obtained from the WASMOD-M runs with the previously defined
250 parameter-value sets. These runoff time series were then used
to calibrate the best wave velocity for each routing algorithm. The
calibration started with a wide range of velocities. Each velocity
was used for each of the 250 runoff time series to derive the NRF
and the K parameter for LRR in order to obtain 250 model efficiencies at the downstream Boluo station for both methods. The wave
velocity that gave the highest efficiency was chosen for each resolution. Runoff-generation parameters were then calibrated with
the efficiency criterion for each fixed wave velocity, and the top
10% parameter-value sets were chosen as behavioural (in the GLUE
sense, see Beven and Binley (1992)).
The best calibrated flow velocities for all three routing methods
(VLRR, V45, and VBEN) were more or less independent of spatial resolution (Fig. 6). The best velocities were always found in a narrow
range, within 0.3–0.6 m/s for the LRR and benchmark routing (VLRR
and VBEN) and within 7–9 m/s for the NRF routing (V45). The final
velocities were taken as VLRR = 0.4 m/s, V45 = 8 m/s and
VBEN = 0.45 m/s (Fig. 6). VLRR was expected to be slower than VBEN
because it had to compensate the earlier arrived flood peaks produced by the LRR method. The stability of V45 was expected since
it was not influenced by a changing network topology, while the
stabilities in VLRR and VBEN were less expected. The fact that the
flow velocities were reasonably scale-independent for all methods
simplified the performance comparison across scales for all three
methods.
When flow networks are up-scaled, the spatial time-delay distribution gets smoothed out. The total delay is also biased by a network meander factor, i.e., a quotient of the ‘‘true” river length, in
our case taken from the HYDRO1k network flow-path length, and
the distance between two reference points in any of the up-scaled
flow networks. The meander factor can influence the optimal
velocity across spatial resolutions. One advantage of NSA is that
it provides a simple scaling relationship for basin parameters at
up-scaled resolutions (Fekete et al., 2001). For example, decreased
river lengths at coarser resolution can be deduced with a simple
scale relation. A linear regression between up-scaled lengths at
all spatial resolutions and ‘‘true” 1-km-pixel lengths gave a slope
of 1.2. This slope was our NSA-derived meander factor for the Dongjiang basin. As expected, it was slightly smaller than the meander
factor, i.e., the quotient of the actual (published) river length to the
idealized river length, of 1.4 derived by Oki et al. (1999).
Results
Scale dependency of routing dynamics
Routing dynamics differed considerably between the LRR and
diffusion-wave-approximation methods when tested on a 30-km
cell. Wave crests got higher and attenuation smaller when LRR
was applied on reaches shorter than a whole cell. It was expected
that, with the postulated wave velocity of 1.44 km/h, the crest
should arrive at the downstream side of the cell after about 20 h.
These 20 h were used by the LRR method as its mean-residencetime parameter K, but the LRR crests arrived after approximately
12 h for a 1-h unit-volume input (Fig. 5a). The diffusion-wave
approximation, on the other hand, modelled crest arrival time at
20 h for small values of the diffusion-coefficient. The response
functions got positively skewed for large D values (Fig. 5a), but
the centre-of-mass arrival times were still around 20 h. The centre-of-mass arrival times for LRR were always significantly smaller
than 20 h. The two methods showed similar behaviour when fed by
a constant 24-h unit-volume input (Fig. 5b). The LRR crest always
arrived earlier than the mean residence time whereas the diffusion-wave crest arrived ‘‘on time”.
The integrated daily response functions show the delay behaviour for the simulation time step used in the water-balance model.
Three results stick out:
(1) Although most discharge arrived on the first 2 days for both
methods, almost half arrived on day 1 with LRR, whereas the
peak flow occurred on day 2 for the diffusion-wave approx-
Routing performance at different spatial resolutions
The routing methods performed very differently for the Dongjiang basin (Fig. 7). The performance was similar between all three
methods at the 50 resolution (9 km cell size). The performance
of the LRR routing decreased linearly with a zigzag pattern when
moving from finer to coarser scales. The benchmark routing had
a slower performance decrease and a similar zigzag pattern. The
NRF routing performed equally well at all resolutions. The NRF
routing was apparently scale-independent both theoretically and
practically. A comparison between the LRR and the benchmark
routings indicated that the linear-storage-release function performed increasingly poorly for coarser scales (Fig. 7). It also proved
that both methods based on aggregated cell information showed a
245
L. Gong et al. / Journal of Hydrology 368 (2009) 237–250
Network response
0.1
(a)
0.08
0.06
0.04
0.02
Network response
0
(b)
0.04
0.03
0.02
0.01
0
Percentage
80
24
48
72
Hours
1 section
2 sections
3 sections
4 sections
5 sections
6 sections
7 sections
8 sections
9 sections
10 sections
(c)
60
40
96
120
144
D=0 km2 / hour
(d)
2
D=1 km / hour
D=2 km2 / hour
2
D=3 km / hour
D=4 km2 / hour
D=5 km2 / hour
D=6 km2 / hour
D=7 km2 / hour
D=8 km2 / hour
D=9 km2 / hour
2
D=10 km / hour
20
0
1
2
3
4
5
Days
6
1
2
3
4
5
6
Days
Fig. 5. Response functions for a 1-h (a) and a 24-h (b) unit-volume input to a 30-km cell produced by a linear-reservoir-routing (LRR) method with 1–10 sub-sections (more
sub-sections result in higher crest; dashed lines) and by a diffusion-wave approximation with diffusion-coefficient values of 1–10 km2/h (solid lines). The zero-diffusioncoefficient signifies a mathematically very small number. Percentage of daily discharge, integrated from the 24-h response functions by (c) LRR and (d) diffusion-wave
approximation.
worse performance when gradually losing hydrographical
information.
Another difference between the routing methods could be seen
in the simulated Boluo hydrographs and posterior parameter distributions produced by the behaviour models (Fig. 8). The result
from year 1983 is presented for illustrative purposes. The NRF
method helped to maintain similar posterior cumulative density
functions for the three WASMOD-M parameters over all spatial
resolutions. The LRR methods were unable to achieve this. The best
simulated hydrographs for the NRF method, which were close to
the observed discharge, consequently showed a very narrow
uncertainty range over all spatial scales.
Limitations of aggregated networks
There was no systematic difference between the performance
for the aggregated network methods (LRR and benchmark NRF)
when driven by spatially distributed and uniform climate input.
The NRF method, on the other hand, seemed capable of using the
extra information in the distributed climate data and its efficiency
was almost equally increased at all resolutions (Fig. 7).
The zigzag pattern for the two network-based methods was a
pure network effect (Fig. 9). The network topology was subject to
a periodically appearing threshold effect at certain length scales.
The reference point at the Boluo station was relocated from one
cell to another at each of these thresholds. This changed the flow
net for the whole downstream part of the basin with a considerable
effect on model performance.
Discussion and conclusions
We could not falsify the hypothesis that storage-based cell-tocell-routing algorithms are less and less suitable as the spatial scale
gets larger and larger but a comparison between a LRR algorithm
and a diffusion-wave algorithm showed that the inherent LRR
assumption of instantaneous reaction to an input at the up-stream
cell border created too early arrival at the outlet. This assumption
is equivalent to neglecting the convective delay in a reach. The delay problem grew with cell size and became significant for sizes
used in many global water-balance models. The diffusion-wave
algorithm, on the other hand, gave delay times that were more
consistent with real-world travel times. This algorithm could be
drastically simplified by assuming near-zero diffusivity when applied in large-scale, global models. This simplification was quite
good for small diffusivities and, when compared to LRR, showed
much better performance for the medium-sized Dongjiang basin.
The NRF routing presented in this study shares some assumptions with existing large-scale routing algorithms. The most important one is that wave velocity does not change as a function of
discharge. When only the down-most gauging station is chosen
as reference point, the method furthermore collapses to a simple
source-to-sink algorithm, which is good enough for most largescale models. The new method is flexible in many ways. Here we
demonstrated it on a daily time step, but application on longer
or shorter time steps can easily be done by recalculating the
parameters t and p at source resolution and re-aggregating them
to a lower spatial resolution for varying time steps.
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L. Gong et al. / Journal of Hydrology 368 (2009) 237–250
1
V = 0.4 m/s
0.9
All other test V from [0.3, 0.6] m/s
0.8
0.7
0.6
5
10
15
20
25
30
35
40
45
50
55
60
30
35
40
45
50
55
60
30
35
40
45
50
55
60
0.89
V = 8 m/s
45
All other testV
Efficency
0.88
45
from [7, 9] m/s
0.87
0.86
5
10
15
20
25
1
0.9
0.8
V = 0.45 m/s
0.7
All other test V from [0.3, 0.6] m/ s
5
10
15
20
25
Cell size in arc-minutes
Fig. 6. Model efficiencies for different velocities at 56 different spatial resolutions (cell sizes varying from 50 to 600 in steps of 10 ) and three routing models: linear-reservoir
model (top), NRF model based on 1-km-resolution data (middle), and benchmark routing with input data for each resolution (bottom).
0.9
0.85
Efficiency
0.8
0.75
0.7
NRF,V
45
= 8 m/s
NRFuni, V 45 = 8 m/s
LRR, V = 0.4 m/s
LRR , V= 0.4 m/s
uni
0.65
Bench, V = 0.45 m/s
Bench , V = 0.45 m/s
uni
10
20
30
40
50
60
Cell size in arc-minutes
Fig. 7. Model efficiencies for three routing models at spatial resolutions varying
from 50 to 600 in steps of 10 . The models are NRF: Network-response function, LRR:
linear-reservoir routing, and Bench: benchmark model using NRF with averaged
hydrographic data from cells at each aggregated resolution. Each model was
calibrated to one flow velocity (V45 and V) for all scales and each model was run
with spatially distributed as well as uniform (subscript ‘‘uni”) climate-input data.
The small meandering factor indicated that the average change
in flow-path length during up-scaling, and thus the change in average delay for a given stream velocity, was small. Changes in individual flow-path lengths and individual delays could still be
large. The wave velocity should scale with the meandering factor
1.2 but other information-aggregation factors evened out this effect. The thresholds in LRR efficiency at certain length scales
(Fig. 7) were shown to be a pure network effect (Fig. 9). This could
impose limitations in the use of low-resolution flow networks for
LRR routing. Routing efficiency is sensitive to flow-path topology,
primarily determined by the relative position of the downstream
station. A given river basin will be assigned a good or bad downstream flow-path topology by chance when global-scale grids are
constructed on the latitude–longitude system. This could influence
global-scale routing result and complicate performance
comparisons.
The strength of the NRF algorithm was clearly shown as a high
and constant efficiency across all studied spatial resolutions
whereas the algorithms based on spatially averaged properties
got impaired performance at coarser scales (Fig. 7). These latter depended on a gradually simplified flow network and led to biased
delay dynamics. This indicated the importance of the spatially distributed delay even if the spatially averaged delay did not change
much over different resolutions. The network-based models did
not perform better, even at a spatial resolution of 50 than the NRF
model in spite of their much higher computational cost.
When moving towards larger cell sizes, the spatial variability in
the input climate field decreased and as a result, the spatial variability of WASMOD-M-generated runoff also decreased. The influence on model performance of spatial resolution in the climate
input was easier to analyse in the NRF case since the delay dynamics was preserved over all spatial scales. It was shown that the network-based models performed equally good/bad with uniform and
distributed climate input whereas the NRF model could take
advantage of the extra information in the distributed field, even
though the improvement was small. The representation of spatially
distributed delay might thus be more important than spatially distributed climate as long as discharge is the main performance indicator. Since a main objective for most global-scale water-balance
models is to reproduce discharge at large river outlets, a correct
and accurate delay distribution should be a high priority.
This study provided some hints for future global water-balance
modelling. It would be natural to increase the spatial resolution of
such models especially when better satellite data become available.
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L. Gong et al. / Journal of Hydrology 368 (2009) 237–250
a
7000
5000
3
Discharge (m /s)
6000
4000
3000
2000
1000
0
J
F
M
A
M
J
J
1983
S
O
1
0.8
0.8
0.8
0.6
0.6
0.6
F(x)
0.4
0.4
0.4
0.2
0.2
0.2
0
0.96
0.98
N
D
J
F(x)
1
F(x)
1
0
0.94
1
0
0
1
A
2
c
4
b
A
3
0
2
4
c
−3
x 10
1
6
−4
x 10
2
7000
Discharge (m3/s)
6000
5000
4000
3000
2000
1000
0
J
F
M
A
M
J
J
1983
A
S
O
1
0.8
0.8
0.8
0.6
0.6
0.6
F(x)
0.4
0.4
0.4
0.2
0.2
0.2
0
0.94
0
0.96
0.98
1
D
J
F(x)
1
F(x)
1
N
0
0
A
4
1
2
c
1
3
−3
x 10
0
2
4
c
2
6
−4
x 10
Fig. 8. Observed (black line) and modelled discharge (gray lines) in 1983 for the Dongjiang basin at Boluo. Runoff generation was modelled with WASMOD-M and routed by
linear reservoir method (a) and aggregated response functions (b). The gray lines represent the best simulation for each of the spatial resolutions ranging from 50 to 600 in
steps of 10 . The three graphs below each hydrograph show the posterior cumulative density functions F(x) for the three WASMOD-M parameters after 250 Monte-Carlo
simulations at each of the 56 spatial resolutions.
Knowledge transfer between scales is always a difficult task, however, both because physical processes depend on scale and because
modellers sometimes use scale-inconsistent approaches. Yildiz
and Barros (2005) point out that simulated runoff generation is con-
siderably deteriorated when flow-network resolution decreases
from 1-km to 5-km while other input data are the same. Our study
provided a simple solution to the problem of Yildiz and Barros
(2005) as shown by the scale independence of the
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L. Gong et al. / Journal of Hydrology 368 (2009) 237–250
Fig. 9. Flow-net representations for the Dongjiang (East river) basin at four spatial resolutions differing pair-wise by 10 . The original 1-km flow net was registered on the
down-most discharge gauging station in Boluo (black circle). Note the effect on the flow-net topology when the Boluo station is relocated from one cell to another because of
the small change in spatial resolution.
posterior distribution of runoff model-parameter values (Fig. 8). This
scale independence indicated that runoff-generation models might
use the same parameter values across scales and that both runoff
generation and discharge dynamics will be scale-invariant. Another
reason behind the scale independence of the runoff-generation
parameters is that WASMOD-M is only driven by climate input.
The value of high-resolution climate input may not be evident when
spatially distributed runoff is aggregated to downstream discharge.
This is illustrated by the fact that our model efficiency was insensitive to climate-input resolution (Fig. 7). However, if scale-sensitive
topographic data (slope, upslope contributing area, etc.) are used,
it could be necessary to combine the proposed routing algorithm
with a scale-invariant runoff-generation algorithm (Pradhan et al.,
2006) in order to transfer parameter values across scales.
Acknowledgements
This work was funded by the Swedish Research Council
grants 629-2002-287 and 621-2002-4352, grant 214-2005-911
from the Swedish Research Council for Environment, Agricultural
Sciences and Spatial Planning, grant SWE-2005-296 from the
Swedish International Development Cooperation Agency Department for Research Cooperation, SAREC, and grant CUHK4627/
05H from the Research Grants Council of the Hong Kong. Parts
of the computations were performed on UPPMAX resources under Project p2006015. We are grateful to Prof Yongqin David
Chen of the Chinese University of Hong Kong for providing the
hydrological data. We also thank Prof Keith Beven for kindly
reading both original and revised manuscript, his valuable com-
L. Gong et al. / Journal of Hydrology 368 (2009) 237–250
ments and corrections greatly improved the language and quality of this paper.
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