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How and to what extent does the spatial and temporal
discretization schema affect GIS-based hydrological modelling?
Honglin Zhu
Department of Geography
Hong Kong Baptist University
Hong Kong SAR, China
20482787@life.hkbu.edu.hk
Qiming Zhou
Department of Geography
Hong Kong Baptist University
Hong Kong SAR, China
qiming@hkbu.edu.hk
ABSTRACT
The justification of the spatial and temporal discretization schema
is a critical step in the development of numerical hydrological
models. Currently, the challenge remains in balancing the error and
uncertainty induced by the algorithm and the mass calculation
caused by the increase of the division of computational units. Thus,
it is necessary to investigate an appropriate discretization scheme,
which not only adequately represents the spatial heterogeneity
characteristics, but also maintains a sufficiently high computational
efficiency, with the constraints of the data validity and availability.
This poster paper proposed a numerical hydrological model using
different spatial and temporal discretization schema. Results show
that the running time revealed an increase by an order of magnitude
with the refinement of the grid size. The results also show that that
the discretization schema impose various influences on different
hydrological processes. For the infiltration process, the effect of the
spatial and temporal resolution depend on the soil type; for the
runoff process, the amount of the runoff was less affected but the
time to runoff was significantly influenced. Establishing a
standardized method to optimize the range of the spatial-temporal
resolution for different the models and environmental scenarios,
however, still remains challenge and is the future investigations.
CCS CONCEPTS
Information systems spatial temporal systems
KEYWORDS
Spatial and temporal resolution, discretization schema, numerical
simulation, discretization error
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SIGSPATIAL '21, November 2 5, 2021, Beijing, China
© 2021 Association for Computing Machinery.
ACM ISBN 978-1-4503-8664-
https://doi.org/10.1145/3474717.3484268
ACM Reference format:
Honglin Zhu and Qiming Zhou. 2021. How and to what extent does the
spatial and temporal discretization schema affect GIS-based hydrological
modelling? In SIGSPATIAL ACM SIGSPATIAL International
Conference on Advances in Geographic Information Systems, November
02 05, 2021, Beijing, China. ACM, New York, NY, USA, 4 pages.
https://doi.org/10.1145/1122445.1122456
1 Introduction
GIS-based hydrological models are widely used tools to simulate
the hydrological response processes such as infiltration and runoff
generation [1-3]. In such models, the determination of the
discretization schema is critical. Generally speaking, the finer the
resolution, the more accurate the simulation results will be [4].
However, Qiu et at. [5] and Politi et al. [6] reported that the
accuracy of the simulation results with the three spatial resolution
were not much different. Thus, it remains inconclusive whether
finer resolution would achieve better modelling results in the
numerical experiments with various conditions. Besides, the spatial
and temporal discretization schema also have effects on the
computational efficiency as different resolution may change the
structure of the model and the number of iterations [7].
In addition, the choice of spatial and temporal resolution have
different effects on the simulation depending on the type and nature
of the hydrological response processes [8-9]. For the infiltration
process, for example, it was found that the parameters related to
soil water movement and the estimated infiltration amount at
different resolution differed by several orders of magnitude [10].
For the runoff generation process, Unami et al. [11] reported that
the simulated peak amount of the runoff was smoothed as the
catchment is divided into fewer and larger sub-basins. It is also
found that the spatiotemporal resolution is more sensitive to the soil
and vegetation conditions, which exhibit more local variations,
compared with the climatic conditions [12-13].
Therefore, our objectives are (1) to conduct a numerical
hydrological model and investigate an appropriate discretization
scheme with high computational efficiency and modelling stability;
(2) to investigate the impact of different spatial and temporal
discretization schema on the simulation results of various
hydrological response process based on the numerical experiments.
In this study, we adjust the temporal resolution and the grid size in
the iterative process to obtain the optimal range under different
05, 2021, Beijing, China Trovato and Tobin, et al.
initial conditions, and quantify the influence of different spatial and
temporal discretization schema on the simulation results.
2 Description of the numerical model
The hydrological process of infiltration and runoff were coupled
in two main modules. The model has been conducted with the
MATLAB programming language. The computer configuration
included: the system type is Windows 10 64-bit operating system,
the processor is Intel core i3 with 3.1GHz, and the installed RAM
is 8G.
Lighthill and Whitham [14] proposed the kinematic wave
equation as an approximate method of Saint-Venant's equation to
estimate one-dimensional surface flow. For shallow surface flow,
ignoring the velocity and pressure head gradient in the momentum
equation [15-16], the first-order hyperbolic partial differential
equation can be obtained:
e
AQ
ib
tx
(1)
Where Ais the area of the cross section; b is the width of the cross
section; Q is the flow rate;
e
i
is the excess rainfall (impermeable
part); t is the time; x is the downslope distance.
Substituting
A b h
and
Q q b
into the equation (1), we can
get:
e
b h q b
i b
t x
2
For turbulent flow conditions, the slope roughness cannot be
ignored, the Manning formula is used to express the relationship
between and. Here it is assumed that the hydraulic gradient of the
slope flow is equal to the angle of the slope, that is, the loss of
runoff caused by surface filling is also considered:
5
3
0
1
( )
s
q S h d
n
3
Where
0
S
is the slope angle,
n
is the roughness, and
s
d
is the
amount of landfill savings.
The infiltration equation coupled with the hydrodynamic model
was obtained by Mein and Larson [14]. Taking into account the
influence of surface water, this equation was chosen considering
the robustness and respect the data availability:
(t)
(t ) (t t) F ln( )
(tt)
(t)
k t ln(1 ) k t
(tt)
sat sat
F
FF F
F
F
(4)
Where F is the cumulative infiltration capacity,
sat
K
is the saturated
hydraulic conductivity of the soil,
()
fsi
,
f
is the soil
water suction at the wet front, and
s
,
0
is the saturated water
content and the initial water content of the soil, respectively.
The second-order Newton method was used to determine the
cumulative infiltration volume within the time increment
t
:
2
00
20
0
0
2ln(1 )
sat
F F
F F
F F
F
F F F
K t F
(5)
The numerical solution of the first-order partial differential
equation is:
1
1
11
1
2
2
2
jj
j
kkkk
k
jkk
jj
keke k
k
h h fw fw
q
t
fw fw
qii
xfw
(6)
To get the numerical solution, a fully implicit schema was used to
solve the Green-Ampt equation and the runoff module was solved
with an explicit schema.
3 The input and designing of the numerical
experiment
A homogenous and rectangular catchment was selected as the study
area. This small catchment is 100 m long and 10 m. Three different
grid sizes of 20cm, 10cm, and 5cm are selected; three different
temporal resolution of 1.44min, 14.4min and 144min were
performed in the numerical model. The input data and the value of
the parameters in the model are presented in table 1.
Table 1: Values of model parameters and input data
Input and parameter Units Value
Temporal resolution min 1.44, 14.4, 144
Grid size cm 5, 10, 20
Rainfall intensity mm/h 1,10,20,100
Soil bulk density g/cm³ High, medium, low
Saturated hydraulic
conductivity cm/h 3.54,4.14,5.74
Initial suction kPa 10 20 30
How and to what extent does the spatial and temporal
discretization schema affect GIS-based hydrological modelling? 05, 2021, Beijing, China
Manning coefficient s/m 0.25
slope 20,30,40
Depression storage and
intercepting mm 1
As for the soil hydraulic properties, three different soil hydraulic
characteristic curves with different bulk density were used to drive
the model, as shown in Figure 1, which is proposed with reference
to the experimental data of Genuchten et al. [17].
Figure 1: Soil hydraulic characteristic curve of with different
bulk density (LC is the low bulk density soil, MC is the medium
bulk density soil, and UP is the high bulk density soil).
4 Results and discussion
As shown in table 2, the running time with different spatial
resolution are presented, indicating that with the refinement of the
grid size (from 20cm to 5cm), the running time shows an increase
by an order of magnitude, which suggest that the determination of
the grid size is of great significance to the computational efficiency.
Table 2: Running time and the estimated accumulative runoff
of model under different spatial resolution
Grid size
(cm)
Running time
(s)
Accumulative runoff
(mm)
20 74.28 12.28
10 512.23 12.435
5 3251.75 13.47
For the estimated accumulative runoff in table 2, the numerical
solutions under three different spatial resolution are stable and there
is no discretization oscillation. When the grid size become finer,
the simulated runoff increased gently. The discharge hydrograph
with three grid size are as shown in the Figure 2. At three different
grid resolution, the peak flow and the streamflow shape withdrawal
consistent trend curve, but with different grid size the runoff
amount is different.
Figure 2 the discharge hydrograph of the runoff simulation
results under three different spatial resolution.
As shown in Figure 3, the simulation of time to runoff can be
affected by many factors, including the environmental factors, such
as rainfall intensity, soil properties, slope and soil moisture content,
and factors related to the model itself, such as the discretization
schema, the equations used in the model and so on. When the
temporal resolution takes different values from 1.44min to 144min,
the simulation results varies with different soil bulk density. For
low bulk density soil, the change of the temporal resolution does
not have much effect on the simulation of time to runoff, and the
estimated response time is almost the same. However, for the low
and medium bulk density soils, when the temporal resolution
increased to 144min, abnormal values were found in the simulation
results, which suggests that the numerical solutions with the
temporal resolution of 144min does not converge.
Figure 3: Simulation of pressure heads with different soil
properties under three different temporal resolution with the
same grid size of 10cm.
Figure 4 illustrates that the infiltration rate computed for different
spatial resolution. For the high bulk density soil, the infiltration
05, 2021, Beijing, China Trovato and Tobin, et al.
rate under different grid size is almost the same at the beginning
and the end of the infiltration. In the middle stage, the estimated
infiltration rate under the schema of 5cm is the largest, and the
rate under the 20cm is the smallest. However, the spatial
resolution had a more significant influence on the estimation
results under the condition with the low bulk density. As shown in
figure 4(b), the simulated infiltration rate obtained with the
resolution of 5 cm is much larger than the one under 20cm
compared with that in figure 4(a).
Figure 4: the infiltration rate under different spatial resolution
with the high bulk density soil (a) and low bulk dentistry soil
(b).
5 Conclusion
This study developed an infiltration-runoff numerical model
coupling the Green-Ampt equation and the kinematic wave
equation. Numerical experiments were implemented using
different spatial and temporal al discretization schema. Is was
found that the running time showed an increase by an order of
magnitude with the refinement of the grid size from 20 to 5 cm.
Results showed the discretization schema impose various
influences on different processes. For the infiltration process, the
effect of the spatial and temporal resolution depend on the soil type.
Under the low bulk density soil with larger hydraulic conductivity,
the simulation results were much more sensitive than under the low
hydraulic conductivity. For the runoff process, the amount of the
runoff was less affected but the time to runoff was significantly
influenced by the spatial and temporal resolution. However, the
challenge remains, to establish a standardized method to optimize
the range of the spatial-temporal resolution for different the models
and environmental scenarios, which is also the subject to our future
investigations.
ACKNOWLEDGMENTS
The research is supported by the Natural Science Foundation of
China (NSFC) General Program (41971386) and Hong Kong
Research Grant Council (RGC) General Research Fund (HKBU
12301820).
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