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Water Resource
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Mesh type runoff model by GIS
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.
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