A novel downstream Flood Hazard Grade Index incorporating upstream Hydrograph Characteristics to predict Debris Flow Runoff

. The July 2020 debris flow in Japan caused enormous damage, and briefing sessions on disaster prevention have prompted demands for detailed explanations and predictions of such phenomena in high-risk areas. It is necessary to obtain four-dimensional risk information, which considers temporal changes in disaster risk, rather than limiting the analysis to conventionally static information. In this study, we developed a method for setting the boundary conditions necessary for debris flow prediction via a four-dimensional hazard map using various types of digital information. To understand the effects of hydrograph characteristics from the upstream, flow discharge was analysed under different flow conditions, such as topography-driven riverbed shear stress, using a one-dimensional numerical model that considers water and sediment flow. Our results suggested that characteristics of the upstream inflow hydrograph affect flood runoff processes downstream; therefore, we developed a separate downstream flood hazard grade index that uses characteristics of the upstream inflow hydrograph as input.


Introduction
With the broad application of detailed threedimensional (3D) model data, such as satellite topographical data [1], further development of models that utilize digital information to predict hazard susceptibility is expected in Japan [2] [3]. For example, landslide disaster countermeasure models offer detailed predictions of disaster risks based on rainfall and landslide forecasting information [4].
A debris flow in Atami City, Shizuoka Prefecture, Japan in July 2020 caused enormous damage [5], resulting in nationwide interest in risks associated with landslides such as debris flows. Subsequent briefing sessions based on disaster prevention hazard maps have prompted demand for detailed explanations and predictions of such phenomena in high-risk areas.
Therefore, to further improve understanding of disaster risks among the general population, we developed the iHazard map project, which includes simple explanations of disaster phenomena using visualizations, as shown in Figure 1. For this purpose, it is necessary to obtain four-dimensional (4D) risk information, which considers temporal changes, rather than limiting the analysis to conventionally static information [6]. In this study, we developed a method for setting the boundary conditions necessary for debris flow prediction, to create 4D hazard maps using various types of digital information.
To understand the effects of upstream hydrograph  [7] was used in this study.
The momentum in the flow direction under depthaverage velocity is calculated as follows: (1) The continuity equation for the total volume of a debris flow is: the continuity equation for the volume of sediment in the debris flow is and the continuity equation for the riverbed level is where u is the average velocity in the flow direction, t is time, x is the flow distance, g is the acceleration due to gravity, τb is the riverbed shear stress, H is the flow surface level (H = z + h), ρ is the interstitial fluid density, h is the flow depth, ib is the erosion/deposition velocity, C is the sediment concentration of the volume flow, C* is the sediment concentration by volume in the movable bed layer, and z is the riverbed height.
Various sediment transport types affected by changes in bed slope are observed in the downstream segments of debris flows. Therefore, the riverbed shear stresses of flow τb were classified into three types according to the sediment characteristics of the debris flow [7], stone debris flow, immature debris flow, and bed load transport, which are represented by the following equations: where ρ is the interstitial fluid density, σ is the bulk the density of sediment, d is the particle diameter of the sediment, and nm is Manning's riverbed roughness coefficient.
The erosion/deposition velocity ib is calculated as : where C∞ is the equilibrium sediment concentration, q is the unit width flow discharge, δe is the coefficient of erosion, and δd is the coefficient of deposition. The equilibrium sediment concentration C∞ for the debris flow is where θw is the water-surface gradient and φ is the internal frictional angle of the sediment.
The generation and development of debris flow are calculated using a staggered scheme and arrangement variables.

Calculation conditions
Debris flow runoff processes affected by upstream inflow hydrograph characteristics were estimated under previously described ideal conditions [7], as shown in Table 1. Flood runoff processes affected by the river width, riverbed gradient, and sediment concentration of the volume flow were analyzed.
To understand the effect of upstream inflow hydorograph characteristics on flood runoff processes, we used numerical models to analyzed seven significantly different hydrograph geometries, as shown in Figure 2.

Factors affected by upstream hydrograph characteristics
To understand the debris flow processes affected by upstream inflow hydrograph characteristics, we analyzed the peak outflow discharge at different observation points (different flow distances) using hydrographs (cases 1-1, 1-2, 1-3, and 1-4) as shown in Figure 2. Other variables were kept constant, i.e., riverbed gradient (i = 0.04), river width (B = 10 m), and sediment concentration of the volume flow (C = 0), as shown in Figure 3. Relationship between flow distance and peak flow discharge for various flow distances. In Figure 3, the y-axis is the non-dimensional peak flow discharge at each observation point (i.e., the peak outflow discharge at each observation point divided by the maximum outflow discharge; Q0max = 40 m 3 s -1 ), and the x-axis is the flow distance between the boundary of the upstream and the observation point.
The peak flow discharge of each hydrograph shown in Figure 2 changed as a result of upstream inflow. To evaluate upstream effects on downstream sediment concentrations, we first focused on floods without sediment. In addition, the results shown in Figure 3 suggest that downstream ratios of peak outflow discharge were nearly unchanged over distances of > 500 m.   Figure 4. Figure 4 shows the relationship between the riverbed gradient and the non-dimensional peak flow discharge. The consideration of upstream inflow hydrograph characteristics altered the peak flow discharge compared to the results shown in Figure 2.
Next, we analyzed peak flow discharge for different river widths, keeping other variables constant, i.e., flow distance from the observation point (L = 1,500 m), riverbed gradient (I = 0.04), and flow sediment concentration (C = 0), as shown in Figure 5. Figure 5 shows the relationship between river width and nondimensional peak flow discharge. Again, the consideration of upstream inflow hydrograph characteristics altered the peak flow discharge compared to the results shown in Figure 2. Finally, flow discharge was analyzed for various flow sediment concentrations, while other variables were kept constant, i.e., flow distance at the observation point (L = 1,500 m), riverbed gradient (i = 0.04), and river width (B = 10 m), as shown in Figure 6. Figure 6 shows the relationship between flow sediment concentration and non-dimensional peak flow discharge. A comparison of Figure 6 with Figures 2-4 indicates that sediment concentrations were not affected by upstream hydrograph characteristics. We assume that these simulation results are representative of the behavior of actual debris flows.  by trial and error using the ratio of the flow discharge difference shown in Figure 3. The resulting relationship between flow discharge and inflow hydrograph characteristics is as follows: , where Qmax is the downstream debris flow discharge, Q0 max is the upstream inflow discharge (Figure 7), and ΣQ85 is the proposed new index obtained through sensitivity analysis, with 15% of the lower flow discharge subtracted from the whole flow discharge (Figure 7), where α and β are correction coefficients. This new index is proposed as a downstream debris flow hazard degree index that considers the upstream inflow hydrograph characteristics, as follows: where FH is the proposed new index, assuming that α = 1.0. If FH is larger, then the debris flow hazard is greater downstream. Our analysis results yielded a correlation coefficient of 0.89, with β = 0.5 (Figure 8). Further study is needed to improve the accuracy of the new index.

Summary
In this study, we used a 1D numerical model to investigate the effects of upstream inflow hydrograph characteristics (e.g., landslide dam installation) on downstream debris flow runoff processes. As a result, we developed a new debris flow hazard index for downstream areas based on upstream inflow hydrograph characteristics.