METHODOLOGY OF A GRID-BASED HYDROLOGICAL
MODEL AND ITS APPLICATION
FEI YUAN
College of Water Resources and Environment, Hohai University, No.1 Xikang Road
Nanjing, 210098, China
Fax: +86-25-83786996; Email address: fyuan@mail.iap.ac.cn
LILIANG REN
College of Water Resources and Environment, Hohai University, No.1 Xikang Road
Nanjing, 210098, China
On the basis of raster digital elevation model data, raster flow vectors, watershed
delineation and spatial topological structure were generated by the Martz and Garbrecht
method for the upper area of Hanzhong station within the Hanjiang River in China. Then,
the Xin’anjiang Model was applied to runoff generation in each grid element with raster
precipitation data interpolated by the step-by-step correction method as the model input.
Finally the Muskingum-Cunge method considering lateral flow into the stream was
applied for flood routing on the raster drainage network. Thus a grid-based hydrological
model was established. The model performs well in terms of the model efficiency
coefficient and of the relative error of flood discharge peak value. The model is not only
able to route the streamflow to the outlet, but also capable of computing conveniently the
hydrograph at any grid within the catchment. As a result, the raster-based digital basin
coupling with the grid-based hydrological model is of theoretical value and of practical
significance to deep understanding of hydrological physical mechanism, water resources
management, and flood prevention within the catchment.
INTRODUCTION
Topography is of great importance to the description, quantification, and interpretation of
land surface processes [1]. Topographical variability not only affects soil information,
plant growth, land-cover and spatial distribution of precipitation, but also controls flow
path and drainage network. Topography is an easily available piece of earth information.
Research over the past decades has demonstrated the feasibility of extracting
topographical information directly from a raster-based Digital Elevation Model (DEM)
[2]. The spatial distribution of land surface characteristics, such as topography, land-cover,
soil, watershed divide, drainage network, catchment area, can be expressed digitally, so
as to avoid using the conventional manual methods [3]. Digital basin derived from DEM
data has become an infrastructure for hydrological research, and provided a solid
technical foundation for the development of grid-based hydrological models.
CONSTRUCTION OF DIGITAL BASIN
Study area
The upper reach of the Hanjiang River, located in Central China, was selected as the
study region. As shown in Figure 1, the area controlled by the Hanzhong streamflow
station is 9,329 km2 and located within 32°35′~ 34°10′ N and 106°10′~ 107°30′ E.
Topography in this area is complicated. Elevation within the watershed ranges from 459
m at the outlet to 3,408 m above mean sea level at the top of the watershed divide.
(a)
(b)
Figure 2. Sketch map showing numerical
arrays of basin boundary (a) and of node
index for each grid within the basin (b)
Figure 1. Raster-based flow vectors in the
upper area of Hanzhong station within the
Hanjiang Catchment
Pre-processing of DEM data
In this study, raster DEM data were obtained from the National Geophysical Data Center
(USA), namely the Global Land One-kilometer Base Elevation (GLOBE) data
(http://www.ngdc.noaa.gov/seg/topo/GLOBE.shtml) at a resolution of 30 seconds.
The Digital Elevation Drainage Network Model (DEDNM) developed by Martz and
Garbrecht [4] was applied for assigning flow-vector directions over flat and depressive
areas of the DEM, and for identifying the stream network and watershed divide. DEDNM
can distinguish and fill two types of depressions: impoundment-depressions and
sink-depressions. As to the flat surface, a relief imposition algorithm [4] is adopted for
flow vector designation, which takes topographical characteristics around the flat area
into account. Then the steepest decent method or the D8 (deterministic eight-neighbor)
method [5], which defines the direction of the steepest downward slope to an adjacent
cell, is used to determine flow vectors over the whole watershed. Figure 1 shows the
raster flow vectors in the upper area of Hanzhong station within the Hanjiang Catchment.
2
Topological relationship of drainage network
Given the row and column coordinates of the outlet grid, DEDNM depicts watershed
boundary automatically, and outputs a raster array defining cells inside and outside the
watershed, where the number ‘1’ represents an inner-watershed grid and ‘0’ indicates an
outer-watershed grid as shown in Figure 2-a. Meanwhile, DEDNM generates a raster
array defining flow vector on each grid cell with eight numbers (1~4 and 6~9)
representing the eight possible flow directions for each studied grid. Since only
sub-catchment-based drainage network can be derived from DEDNM, extraction of
topological relationship for raster-based drainage network necessitates a processing on
the raw outputs from DEDNM. The detailed processing procedures are as follows: (1)
Encode node index for each grid cell within the watershed. Search the raster array of
basin boundary (Figure 2-a.) line by line for the grid cells whose values are ‘1’, and
encode those grid cells in terms of searching sequences. In this study, the upper area of
Hanzhong station is composed of 13, 031 grids and the node index of the outlet grid is
‘10424’; (2) Identify the inflow and outflow nodes for each inner-watershed grid. Search
the raster array of flow vectors for the inflow and outflow nodes, and identify their
corresponding node indices according to the raster array of node indices (Figure 2-b); (3)
Compute routing execution sequences. As shown in Figure 3, all inner-watershed grids
are classified into N levels. Outlet grid belongs to Level 1; inflow grids of Level 1 are
grouped into Level 2; inflow grids of Level 2 are ranked as Level 3; the rest is deduced by
analogy. Assumed that grid number of each level is ki (i=1, 2,…, N), kN grids in Level N
perform routing firstly, whose routing execution sequences are 1, 2,…, kN respectively;
kN-1 grids in Level N-1 route after Level N, whose routing execution sequences are kN +1,
kN +2,…, kN +k N-1 respectively; by analogy, kI-1 grids in Level I-1 compute routing after
Level I, whose routing execution sequences are kN+kN-1+…+kI+1, kN+kN-1+…+kI+2,…,
kN+kN-1+…+kI+kI-1; and outlet grid routes in the end, whose routing execution sequence
is kN+kN-1+…+kI+…+k2+k1.
Table 1. Routing execution sequence and topological relationship of drainage network
Node
Index
Execution
sequence
1
2
3
Inflow index
4
5
6
7
1
2
3
568
567
527
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
3
-1
-1
-1
-1
-1
-1
-1
Next node
downstream
index
7
7
2
7000
7001
31
35
-1
-1
7001
7002
846.1
846.1
0.7195
0.7195
13030
13031
10426
10425
-1
-1
13027
13028
846.1
846.1
0.7195
0.7195
6904 6905
-1
6906
-1
-1
-1
-1
-1
-1
-1
-1
6999 7092 7093
7000
-1 7094
-1
-1
-1
-1
Note: ‘-1’ in Table 1 represents a null datum.
3
-1
-1
Channel Drainage
length
area
/m
/km2
846.1
0.7195
1196.4
0.7195
846.1
0.7195
After the above data processing, raster-based routing execution sequences and
topological relationship of drainage network (Table 1) are generated, which provide a
solid foundation for developing a grid-based hydrological model.
Figure 3. Sketch showing execution
sequences of routing computation in the
grid-based hydrological model
Figure 4. Sketch of the Muskingum-Cunge
flood routing method
MUSKINGUM-CUNGE METHOD
In general, there are two means for streamflow routing in the case of multi-tributaries. One
is called superposition-after-routing means. Streamflow is routed from each tributary to
the outlet of a catchment, respectively. Then a summation is made with respect to the
routed discharge at the outlet, on the supposition that flood waves from tributaries
propagate independently without disturbance each other. It is applicable to river channel
with steep bed slope and fewer disturbances between mainstream and its tributaries.
Muskingum method [6] belongs to this category and is usually employed in streamflow
routing for multi-tributaries river channel. The other is called routing-after-superposition
means. Streamflow is routed from external upstream node to the internal downstream
junction node. At the junction, the discharge is routed to next downstream junction after
the discharge routed from upstream tributaries is added together. The above routing
algorithm continues until river flow is routed to the outlet of a catchment. Compared with
superposition-after-routing means, routing-after-superposition one is more reasonable for
it reflects physical process of flood propagation. Since the raster-based drainage network
is derived from DEM by the steepest decent method, the routing-after- superposition
means with lateral flow considered is implemented in this research, so as to build a digital
river flow routing model.
Basic equations
Cunge [6] applied a four-point explicit finite difference scheme to solving the
Saint-Venant continuity equation and deducing the Muskingum-Cunge Method. In the
finite difference scheme, the space difference weight is equal to 1/2, and the time
4
difference weight x is equivalent to 1/2－D/(C·Δl), where D is the diffusion coefficient,
C is the wave speed, and Δl is the space step [7]. The Muskingum-Cunge equation that
considers lateral flow into the river [8] is as follows:
Q nj11  C1Q nj  C2Q nj1  C3Q nj1  C4
(1)
where C1=(xk＋0.5Δt)／[(1－x)k＋0.5Δt]; C2=(0.5Δt－xk)／[(1－x)k＋0.5Δt];
C3=[(1－x)k－0.5Δt]／[(1－x)k＋0.5Δt]; C4=qΔtΔl／[(1－x)k＋0.5Δt]; k is the
flood travel time over a sub-reach with a length of Δl and is equal to Δl／C; x is the
Muskingum-Cunge weight coefficient; and q is the lateral flow per unit length.
Application of the Muskingum-Cunge method to grid-based drainage network
Based on the D8 method, water in river channel within a grid cell is assumed to flow in
vertical, horizontal or diagonal directions in an aeroview. Runoff produced within each
grid cell flows into river channel in the form of lateral flow. The region as shown in
Figure 4 consists of six grid cells. Streamflows Qa(t), Qb(t) and Qc(t) from a, b and c grids
all flow into e grid. The total discharge into e is Qe1(t), and the outflow from e is Qe2(t). In
order to compute Qe2(t), an application of the Xin’anjiang model [9] is performed to
calculate lateral flows qa(t),qb(t), qc(t) and qe(t) in a, b, c and e grids respectively; then
equation (1) is used to route the outflows Qa(t), Qb(t) and Qc(t) from a, b and c grids and
Qe1(t) is equal to the summation of Qa(t), Qb(t) and Qc(t); finally Qe1(t) is routed by
equation (1) to be Qe2(t), the outflow from e.
In general, grid-based drainage network is multi-branch river network, which is
much more complicated than the network in Figure 4. Flood routing over the grid-based
drainage network is also complicated. According to the routing execution sequences and
topological relationship of grid-based drainage network (Table 1), the Muskingum-Cunge
method is applied to flood routing based on the following procedures: (1) Initialize such
arrays as Sequ[M], Up1[M], Up2[M],…,Up7[M] and Down[M] to store routing execution
sequence, seven inflow node indices and next code downstream index for each grid cell
within the watershed, respectively; (2) Run the Xin’anjiang model to compute the lateral
flow per unit length at each grid cell within the watershed; (3) Set the variable K equal to
1; (4) Search the array Sequ[M] to find a cell indexed as N, whose routing execution
sequence is K. If Up1[M], Up2[M],…,Up7[M] are all equivalent to –1, there is no inflow
into the cell N and outflow at N is only produced by its own lateral flow; if the inflow
cells of N are indexed as L1, L2, …, Li (Li= Upi[N], Upi[N]≠-1, i=1,2, …, 7), inflow
into N is equal to the sum of the outflows from cells L1, L2, …, Li; and (5) Route the
inflow at N to the downstream cell indexed as Down[N] by equation (1). If Down[N] is
equal to –1, flood routing is ended; if Down[N] is unequal to –1, set K equal to K+1, and
go back to procedure (4).
5
CASE STUDY
In this study, twenty flood events (Δt=1h ) at Hanzhong station from 1980 to 1986 were
selected to perform flood simulation. First, the step-by-step correction method [10] was
utilized to interpolate gauge rainfall data into raster grids with the same resolution as the
DEM data; then the Xin’ anjiang Model with the raster rainfall data as its input was
applied for runoff production and overland flow concentration over each cell within the
watershed; finally, the Muskingum-Cunge method was employed in river flow routing.
Model calibration was performed manually and focused on matching the shape of the
hourly hydrograph. By calibration, the Muskingum-Cunge parameters k and x, time step
Δt and space step Δl were assigned such values as 10 min, -0.4, 10 min and 1000 m,
respectively. Table 2 shows flood simulation results at Hanzhong station.
Table 2. Hourly flood discharge at Hanzhong station simulated by the grid-based
hydrological model
Flood Peak
Flood
number
8001
8002
8004
8005
8006
8102
8104
8105
8201
8203
8302
8303
8305
8306
8402
8404
8405
8406
8502
8602
Observed
/ m3·s-1
Calculated
/ m3·s-1
Error
/%
Peak time
error
/h
3240
6040
1860
3320
1950
5980
8310
4620
2200
3460
3110
3500
5260
3370
2900
1760
1600
2800
2400
1390
2559
6925
2030
3808
1848
8228
9749
4475
1729
3860
2278
2498
5649
2810
2889
1516
1114
2411
2205
1339
-27.2
14.7
9.1
14.7
-5.2
37.6
17.3
-3.1
-21.4
11.6
-26.8
-28.6
7.4
-16.6
-0.4
-13.9
-30.4
-13.9
-8.1
-3.6
5
2
-3
-2
-3
-1
2
2
6
2
0
-3
0
2
6
-2
-4
0
3
7
Runoff depth
observed
/ mm
Calculated
/ mm
Error
/%
Model
efficiency
coefficient
32.8
88.9
29.4
41.7
54.4
117.1
405.3
260.4
32.8
142.6
59.5
59.9
114.4
45.3
62.3
27.6
23.7
91.8
81.5
36
29.4
94.1
32.9
51.9
65.0
134.6
436.8
235.0
30.8
153.1
49.5
59.5
120.3
45.0
57.4
28.9
28.2
97.2
83.4
32.3
-10.3
6.0
11.7
24.4
19.5
15.0
7.8
-9.7
-6.3
7.4
-16.7
-0.7
5.1
-0.7
-7.8
4.6
19.0
6.0
2.2
-10.3
0.850
0.933
0.835
0.792
0.786
0.889
0.867
0.941
0.863
0.846
0.845
0.838
0.959
0.930
0.855
0.822
0.874
0.824
0.900
0.781
As shown in Table 2, the grid-based hydrological model has obtained encouraging
results in simulating the hydrographs at Hanzhong station. In 70 percent of flood events,
the flood peak relative error is within the range of ±20%. The peak time error that is within
four hours occupies 80 percent of floods. The relative error of runoff depth in 95% of
floods events is not exceeding ±20%. In terms of the Nash-Sutcliffe model efficiency
coefficient, cases that are greater than 0.7 and 0.8 make up 100 and 85 percent of the total
floods, respectively.
6
With the routing-after-superposition means, the model is not only able to route the
streamflow to the outlet, but also capable of computing conveniently the hydrograph at
any grid cell within the catchment. Figure 5 shows the routed hourly hydrographs for No.
8305 flood at the Hanzhong, Chadianzi, Wuhouzhen and Tiesuoguan stations.
Figure 5. Hourly flow discharge of No. 8305 flood at the Hanzhong, Chadianzi,
Wuhouzhen and Tiesuoguan stations, respectively
CONCLUSIONS
In this study, a grid-based hydrological model was established, where the Xin’anjiang
model was employed in runoff production over each grid cell within the watershed and
the Muskingum-Cunge method was applied to flood routing according to routing
execution sequences derived from DEDNM. A case study shows that the model can
simulate the floods well. Meanwhile, the model is not only able to route the discharge
streamflow to the outlet, but also capable of computing conveniently the hydrograph at
any grid cell within the catchment, which is of theoretical value and of practical
significance to deep understanding of hydrological physical mechanism, water resources
management, and flood prevention within the watershed.
The grid-based hydrological model is indeed a digital hydrological model, which
provides a solid foundation and powerful technical supports to apply remotely-sensed
rainfall data to hydrological models. Radar-or-satellite-captured precipitation data have
an advantage of high spatial and temporal resolutions. Such an advantage may be fully
and completely utilized in the grid-based hydrological model. With a closed combination
of remotely sensed precipitation data and rainfall-runoff models, the puzzling rainfall
input problem in hydrological models can be overcome. Consequently, it will be of great
benefit to enhance flood forecast accuracy and forecast period.
7
However, due to the unavailability of measured data on channel characteristics for
the Hanjiang River, parameters for the Muskingum-Cunge method (x and k) have to be
calibrated manually by matching observed hydrographs, and these parameter values over
all the sub-reaches are set to be uniform, disaccord with spatial heterogeneity of channel
characteristics. Therefore, it is suggested that an application of geographic information
system (GIS) and a hydraulic method should be made to deduce a spatial distribution of
routing parameters directly so as to improve the flood routing performance.
ACKNOWLEDGEMENTS
This work is supported by the National Key Basic Research Development Programme,
Ministry of Science and Technology, the People’s Republic of China under Grant No.
2001CB309404, and the National Natural Science Foundation of China under Grant No.
40171016.
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