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Mesh type runoff model by
GIS
Ryuzo Yokoyama, Luo Xiao Bo
Department of Computer Science I wate University
Morioka, I wate , Japan 020
Department of Computer Science I wate University
Morioka, I wate , Japan 020
Abstract
A mesh type run-off model (MR model) is proposed. A basin is divided into square meshes with an uniform size. The runoff characteristics of each mesh are described as tank models. Each mesh is related to its adjacent meshes under the flow-down connection and the total meshes form a tree graph under the connection, of which root is the mesh at dam site of the run-off volume. The MR model can directly describe the spatial and temporal variations of precipitation and runoff parameters, which observed /estimated by remotely sensed data and derived from a GIS of the basin .The model is applied to Yuta dam basin located in northern Japan and demonstrated promising results.
Introduction
A mathematical model to describe the relationship between a precipitation and a runoff volume in a basin, a runoff model, is one of the most important topics in the hydrology. Various types of runoff models, e.g. tank model, kinematic wave method, unit diagram method, etc, have been popularly used. The run-off is a complicated process depending upon spatial and temporal factors of precipitation and run-off conditions. The mathematical models of those, those, however, have been described by a lumped parameters systems with limited number of state described by a lumped parameters systems with limited number of state variable due to the restricted observation techniques.
In recent years, however, we have gotten new technologies to describe a run-off model more precisely as follows.
- Remote Sensing: Land cover/ use, snow-covered area, topography, clod
distribution can be detected from earth observation satellite.
- Hydrological telemetering system: Various observation techniques for
spatial distribution of rain such as rain telemetering systems on the
ground and train radar systems have been developed.
- Geographic information system: various geographic information
systems have been developed. By applying the techniques, the run- off
models are conveniently developed.
- Advanced computer system: The computer technology has been developed remarkably in recent years. A rather large scale computation of a detail runoff model can be handled by modern computer systems.
Structure of MR model
The conditions related to the run-off dynamics of a basin such as topography, land cover/use, distribution of rain, etc, are not homogeneous but vary spatially and temporarily. MR model is directed to describe those in homogeneities of run-off characteristics. A basin is divided into square meshes with a suitable size. A run-off dynamics of each mesh is to be described by the tank models with one hole or two holes. Each mesh is related to its adjacent mesh existing at the flow direction of the run-off water . Depending upon the topographical condition of meshes, two kinds of tank models are assumed as follows
- Slope mesh: A mesh which has neither rivers nor streams inside is
called a slope mesh. Its run-off dynamics is assumed to be described by
a two-hole tank as shown in Figure 1. The water from the tank is to flow
down to its related mesh. The mathematical description of the run-off is
given by.
Adx (t) /dt = Ar(t) -Qu (t) -qd (t) + qn (t)---------------------(1)
Where
A : area of the mesh,
X(t) : water level of the slope tank at time t,
r(t) : rain rate to the mesh at time t,
qu (t) : run-off rate from the upper hole of the tank at time t,
qd (t) : run-off rate from the lower hole of the tank at time t,
Qa (t) : total flowing-in water rate from the upper related tanks at time t,
h : height of the upper hole in the tank Qu (t) and qd (t) are described to be
qd (t) = ax(t)
where a and b are flow -out resistances of the upper and the lower holes respectively . Both quand qd (t) are to flow down to its adjustment down stream mesh.
- Stream mesh: a mesh which has rivers or streams inside is called a
stream mesh. Its run-off dynamics is assumed to be described as a
composition of a slope tank and a stream tank as shown in figure 2.The
dynamics of the slope tank is same to that of eq. (1) . The dynamics of
the stream tank is given by
ds(t) / dt = q u (t) +q d ((t) - S (t-i) ----------(2)
Where q u (t) and q d (t) are the flow-out rate from the slope tank of own mesh,and the other items are
S(t) : the water volume in the stream tank at time t,
Rn(t) : rate of the water flowing into the stream tank from its upper andjacent atream tanks at time t,
t : the time lag of the stream tank, which is specified by L/W where L is the length of the stream in the mesh, and W is the parameter determined as the mean gradient of the stream following to the Kraven's formula .
The flow direction of each mesh is specified as one of the eight directions (0 ~7) to the adjacent neighbour meshes as shown in Figure
3. For a slope mesh, the flow direction is determined to be that of dominant slopes inside .For the stream mesh; the flow direction is that of the stream. When several streams exist, the flow direction is specified as that of dominant streams. Under the flow-down relation, the total meshes provide a tree graph of which root is the mesh at the run-off observation site
The slope tank has tree parameters of (, B and h, which must be determined specified to describe run-off volumes of a basin relevantly . For the purpose, by referring to the past run-off observation data, those parameters are determined to minimize the error square integral between the observed to minimize the error square integral between the observed runoff and the MR model run-off . At the first stage of the MR model1 development, we assumed those parameters to be uniform for all slope tanks. Subsequently the performance index for the calculations of the optimal parameters are given by
Where V(t) : an observed run-off volume at the observation site at time t,
F(t) : and output run-off from the MR model st time t
Ts : and appropriate time before precipitation when the run-off volume is stable,
Tf : an appropriate time after precipitation when the run-off recovered to be stable.
Yuta dam, which is one of the flood control dams in the Kitakami river network in northern Honshu, Japan as shown in Figure 4. It has the area of 583Km2, and its maximum and minimum elevations are 1500m and 250m, respectively. In the basin, a rain telemetering system with seven ground stations is installed. The run-off volume has been measured at the dam site. The basin was divided into 1 km square meshes. Figure 5 shows the tree graph of the meshes. The geographic information's for the meshes are organized into a file with a two-dimensional multi-layer structure and used as the fundamental data of the MR model as shown in Figure 6.
Simulation of the MR model
Four cases of flood in the past observation data, which were all precipitations in summer as in Table 1, were considered as the examples of the model simulation. The rain rate for each mesh was assumed to the observed rain rate at its nearest ground station. For each run-off observation data, the optimal parameters of ;, B and h under the performance index of eq.(3) were calculated by the iteration method. Figure 7 shows the results of the simulation of the four cases. In each case, the run-off from the MR model efficiently follows to the observed one. This means that the MR model might be excellent when appropriate parameters were applied. The ptimal parameters, however , vary for each rain . In practical applications of the model, unified values of optimal parameters are necessary.
In this paper, the MR model could only consider the spatial characteristics of the precipitation, but it could not consider those of the run-off characteristics since the unified values of parameters were specified to all slope meshes . In the next step, the algorithms to determine optimal run-off parameters depending upon the run-off conditions are necessary to be developed.
Conclusions
The MR model can directly describes the spatial and temporal variation of runoff in a basin well. By applying the model to Yuta dam basin, the results of the simulation of the past rains relevantly realized the observed run-offs and the model could fundamentally demonstrate promising results. The optimal parameters for thefor case studies were unified but changed in a rather wide ranges. In this sense, further investigations are expected to improve the model to have unified parameters for rains.
Table 1: Optimal parameters for the slope meshes of Yuta Dam MR mode 1.
| Case ID | Date | a | b | h |
| A | 1975.7.10 10: 00am | 0.014 | 0.094 | 20.40 |
| B | 1975.8 6 10:00am~8. 9 09 : 00am | 0.003 | 0.028 | 13.38 |
| C | 1980.6.16 10:00am~6.19 09:00am | 0.007 | 0.072 | 22.60 |
| D | 1984.7. 8 10:00am~ 7.11 09:00am | 0.014 | 0.16 | 22.60 |