## 5 Methodology for Flood Analysis
### Iterative Calculation Method to Obtain the Entire Flow Field  
As discussed in Section 4, due to damage to the water gauge at the Kosakuda Observatory during the 2019 flood, the hydrograph data at the upstream end of the calculation are not known. In addition, because floods of this magnitude were unprecedented, the corresponding river channel roughness coefficient is also unknown. Therefore, an iterative procedure was adopted, as described in Figure 23.  
In the figure, the thick blue arrow represents the river channel, and the vertical arrows represent the water exchange between the river channel and floodplains or between the river channel and the HDP. The variables expressed as green symbols were determined in advance from the observation data, the red symbols are the variables that were determined  
Figure 21: Overflow rate calculated from water gauge data during the 2019 flood.  
by the iterative calculation, and the black symbols are the variables that are intermediate results during the iterative calculation. The meanings of \(q_{A},\ q_{B},\ q_{C},\ q_{D},\ q_{E}\) and \(q_{H}\) are the same as those given in Figure 20. The water level at the downstream end of the calculation is equal to the observed water level at Sukagawa, \(H_{S}\) (see Figure 18a). \(Q_{u}\) and \(Q_{d}\) are the river channel discharges at the upstream and downstream ends, respectively, the latter of which must eventually converge to the discharges observed at the Sukagawa Observatory, \(Q_{S}\) (see Figure 18a). \(H_{A}\) and \(H_{B}\) are the water levels at the points of the old road bridges where the 3 L water gauges were installed, and they must eventually converge to the observed records. The variable \(n\) is the roughness coefficient of the river channel, which is determined in reference to the observed water levels at points A and B, as well as the longitudinal profile of the highest water level.  
The iteration procedure was as follows. Steps (1), (2), and (3) below correspond to the numbers shown in blue in Figure 23.  
1. First, an appropriate \(Q_{u}(t)\) for the upstream end boundary condition is assumed, and the downstream boundary condition \(H_{S}(t)\) is set to perform a one-dimensional unsteady flow analysis considering the water exchange with the floodplain and the HDP. In this calculation, \(q_{A}(t),\ q_{B}(t),\ \text{and}\ q_{D}(t)\) are given as the hydrographs shown in Figure 21, while the unknown variables \(q_{C}(t),\ q_{H}(t),\ \text{and}\ q_{E}(t)\) are calculated using Equation (1).
2. The downstream end flow rate \(Q_{d}(t)\) obtained by the calculation is compared with the Sukagawa observed flow rate \(Q_{S}(t),\) and the upstream flow rate is updated by  
Figure 23: Diagram of the iterative flood modeling procedure.  
Figure 22: Calculated maximum inundation depth and the measured inundation area.  
subtracting their difference \(\Delta Q(t-\Delta t)\) from the \(Q_{u}(t)\), in which \(\Delta t\) is the time for floodwater to reach the downstream end from the upstream end.
3. [PERSON]'s roughness coefficient, \(n\), is updated with reference to \(H_{A}(t)\) and \(H_{B}(t)\) obtained by the 3 L water gauges, and the longitudinal distribution of the highest water level obtained from the flood traces.  
First, \(q_{C}(t)\), \(q_{H}(t)\), and \(q_{E}(t)\) were set to zero, and \(Q_{u}(t)\) was assumed as the following equation to perform preliminary calculations to determine \(\Delta\)\(t\).  
\[Q_{u}(t)\Leftarrow Q_{S}(t)+q_{A}(t)+q_{B}(t)+q_{D}(t). \tag{3}\]  
Figure 24 compares the calculated downstream end flow rate and the Sukagawa Observatory flow rate. Because there is a time difference of about 30 min, we fixed \(\Delta t\) to this value in subsequent iterative calculations.