# Development of a Practical Evaluation Method for Tsunami Debris and Its Accumulation
## 2 Practical Method for Evaluating Tsunami Debris Accumulation and Thickness
The fragility function, which follows the form of fragility function for earthquake motion, is defined by Equation (4), based on the normal distribution function \(\Phi\)[19], as follows:  
\[P_{D}(x)=\Phi\bigg{[}\frac{x-\mu}{\sigma}\bigg{]}=\int_{-\infty}^{x}\frac{1}{ \sqrt{2\pi\sigma}}exp\bigg{[}\frac{(t-\mu)^{2}}{2\sigma^{2}}\bigg{]}dt \tag{4}\]  
Here, \(P_{D}\) is the probability of total loss damage, \(x\) represents the external force (here, the tsunami inundation depth), \(t\) is the variable of integration, and \(\mu\) and \(\sigma\) are the average and standard deviation of \(x\); \(\mu\) corresponds to the inundation depth at which the damage probability is 50%. The values of the statistical variables \(\mu\) and \(\sigma\) were determined by the conventional method [19]. A via coast, such as the Sanriku coast, is characterized by relatively steep slopes. Therefore, the building damage caused by tsunamis was expected to be complicated by the return flow of the tsunami waves, including reflected and superposed waves, due to the steep topography at the edges of the inundated areas. In contrast, on a coastal plain with a straight coastline, most building damage is caused by the tsunami run-up flow, and the influence of the return flow and reflected or superposed waves is assumed to be very weak. The building fragility function was constructed for a plain coast with a straight shoreline, where the complexity of the tsunami flow field is considered to be low, and a clear relationship exists between tsunami magnitude and building damage. In this study, we used data of building damage caused by the 2011 Tohoku earthquake tsunami from cities on the Sendai plain and other coastal plain areas struck by the tsunami to construct the tsunami-induced building fragility function: Ishinomaki City (Ishinomaki Plain only), Higashi-matsushima City, Tagajo City, Sendai City, Natori City, Iwanuma City, Watari Town, Yamamoto Town in Miyagi Prefecture, and Soma City, Minami-soma City, and Iwaki City in Fukushima Prefecture (Figure 10). Data used included the proportion of wooden buildings among the damaged structures and tsunami inundation depth. In Equation (4), \(\mu\) and \(\sigma\) depend on regional characteristics; therefore, we investigated the dependence of \(\mu\) and \(\sigma\) on the proportion of robust buildings and the typical topographic slope in the analysis area. Here, reinforced concrete and steel buildings are defined as robust buildings, and \(Mr\) is defined as the proportion of robust buildings among all buildings in the inundation area. To determine the representative topographic slope, \(i_{\rm F}\), the inundated area was divided into sections about 0.5-1 km long in the alongshore direction; then, in each section, the slope in the direction perpendicular to the shoreline was calculated, and the average slope in all sections was used as \(i_{\rm F}\). We examined the relationships between \(Mr\) and the coefficients \(\mu\) and \(\sigma\), and between \(i_{\rm F}\) and the coefficients. We found no correlation between \(\mu\) and \(Mr\), but \(\sigma\) tended to decrease as \(Mr\) increased (Figure 11). These trends were the same, regardless of whether \(Mr\) represented buildings that had been totally destroyed or ones that had been washed away. We also found no correlation between \(\mu\) and \(i_{\rm F}\), although \(\sigma\) tended to increase as the slope of the terrain increased (Figure 12), whether the totally destroyed or the washed away buildings were considered. Therefore, because we could not confirm any dependence of \(\mu\) on \(i_{\rm F}\) or \(Mr\) within the coverage area of this study, we used a constant value for each damage type. In the case of totally destroyed buildings we used \(\mu\) = 2.26 \(\pm\) 0.51 and we adopted \(\mu\) = 1.75 for safe-side evaluation. In considering those buildings that had been washed away, we used \(\mu\) = 3.57 \(\pm\) 0.52 and adopted \(\mu\) = 3.05 for safe-side evaluation.