Loss Functions for Small Marine Vessels Based on Survey Data and Numerical Simulation of the 2011 Great East Japan Tsunami

The vulnerability of marine vessels were reviewed using information on four tsunami events in Japan, namely, the 1896 Meiji-Sanriku tsunami, the 1933 Showa-Sanriku tsunami, the 1983 Japan Sea tsunami, and the 1993 Okushiri Island tsunami. The locations of these events are shown in Fig. 1. Shuto (1987) summarized the boat and ship damage that resulted from the tsunamis in 1896, 1933, and 1983. The damage data for the 1896 event provided information on whether vessels survived, were damaged, or were washed away. The term washed away was applied to vessels that were washed from their moorings and recovered somewhere else or were missing. Most of the vessels were nonmotorized or small motorized vessels. The ratio ($D_1$) of the number of vessels damaged or washed away to the total number of vessels in each port was determined using Eq. (1)where $A$ = number of vessels that were washed away; $B$ = number of vessels that were damaged; and $C$ = number of vessels that survived. The results were plotted against the reported maximum tsunami height for each village [Fig. 2(a)]. The damage data for the 1933 event provided information on two types of vessels: small vessels without motors [Fig. 2(a)] and large vessels with motors [Fig. 2(b)]. The damage ratio formula [Eq. (1)] was used to plot the damage ratios for the 1933 event in the same way as for the 1896 event.

For the 1983 event, a weighted damage ratio ($D_2$) was calculated using the following:where $a$ = number of vessels that were washed away; $b$ = number of vessels that had major damage; $c$ = number of vessels that had moderate damage; $d$ = number of vessels that had minor damage; and $e$ = number of vessels that were not damaged. The vessels were classified as follows: vessels without motors, vessel weighing less than 5 t, and vessel weighing 5–10 t. The weighted damage ratio is plotted with respect to the measured maximum tsunami height for each village in Figs. 2(a and b). Aketa et al. (1994) used the same definitions to apply the damage ratio ($D_2$) to data for vessel damage during the 1993 tsunami [Fig. 2(a)].

Based on the damage ratios calculated for the previous four tsunami events mentioned, it was concluded that damage was observed for some small vessels when the tsunami height was more than 2 m, and that damage was observed for all small vessels when the tsunami height was more than 5 m. However, half of the large vessels had no damage even when the tsunami height was more than 5 m. Not enough data are available to determine damage ratios for large vessels for tsunami heights more than 10 m.

This section briefly describes the types of damage that tsunamis cause to marine vessels in terms of some tsunami parameters (e.g., tsunami height and flow velocity) based on experience from past tsunami events and from experiments.

Linear regression analysis and normal or log-normal distributions have been widely used to derive fragility functions that express the relationship between tsunami characteristics and the probability of damage. These tools have been applied mainly to the analysis of damage data from the 2004 Indian Ocean tsunami, including building damage in Indonesia (Koshimura et al. 2009), Thailand (Suppasri et al. 2011), and Sri Lanka (Murao and Nakazato 2010); bridge damage in Indonesia and Sri Lanka (Shoji and Moriyama 2007); and damage to mangrove trees in Thailand (Yanagisawa et al. 2009) and Indonesia (Yanagisawa et al. 2010). Similarly, they have been applied to the 2011 tsunami to estimate the damage to buildings (Suppasri et al. 2012, 2013), pedestrian bridges (Muhari et al. 2012), road and highway bridges (Shoji 2013), and breakwaters [Port and Airport Research Institute (PARI) 2013]. In this study, the fragility function concept was applied to the development of loss functions because the previously mentioned studies used data on damage to structures to calculate damage probabilities, whereas this study used economic data on vessel damage and loss.

For a collaborative research study by Tohoku University and the Willis Research Network, conducted as part of the Pan-Asian/Oceanian tsunami risk modeling and mapping project, data were collected on 20,134 damaged small marine vessels, including information on the registered location (port name), tonnage, motor type, material, damage type, and insurance. Fig. 3 shows the data distributions. Approximately 99% of the vessels in this present study were in the ports in which they were registered [Fig. 3(a)], but the exact locations in the ports were unknown. To address this limitation, it was assumed that all of the vessels were in the location at which they were registered during the tsunami. As is the case for buildings (higher buildings are more resilient than smaller buildings because of their intrinsically stronger structural makeup), it was expected that larger and heavier vessels would have less damage than smaller vessels. Therefore, the vessels were first classified by tonnage.

Approximately 92% of the vessels weighed less than 5 t, which can be considered the weight limit for small vessels. Small vessels were further divided by motor type (outboard or inboard). Fig. 3(b) shows that approximately 71% of the vessels weighing less than 5 t had outboard motors and 21% had inboard motors. An outboard (exposed) motor is more vulnerable to potential wave damage, which impairs the ability of the boat to move to safety. In general, vessels weighing less than 5 t are of the fiber-reinforced plastic (FRP) type, and vessels weighing more than 5 t are constructed of aluminum or steel. A loss ratio was calculated for each vessel, defined as the actual insurance payment divided by the insured value, to quantify the damage level.

The four main causes of vessel damage reported are shown in Fig. 3(c). Approximately 22% of the reported vessel damage was directly related to the tsunami, (i.e., sinking or overturning of vessels).

The data were then categorized into three types according to the characteristics of the tsunami height and the arrival time: (1) the whole region, (2) regions near the tsunami source, and (3) regions far from the source of the tsunami. Iwate, Miyagi, and Fukushima were considered to be regions near the source of the tsunami, because they were hit by tsunamis more than 10 m in height that arrived within 1 hour, leaving little time for fishermen to take action in response. However, the tsunami arrived at Hokkaido, Aomori, Ibaraki, and Chiba more than 1 hour later, which increased the time available for the fishermen to respond, and the reported tsunami heights at those locations were less than 10 m.

Approximately 80–90% of the data corresponded to 100% losses, according to the damage criteria of the data set because stranded vessels on land were mostly considered 100% losses. Fig. 4 shows examples of damaged marine vessels that were considered 100% losses. The loss ratio range of 90.1–100% was set as damage level 10 and 10% increments of loss ratio ranges below 90.1% were set as damage levels less than 10. For example, the loss ratio range of 0–10% was set as damage level 1, and the loss ratio range of 10.1–20% was set as damage level 2. Although damage level 10 should be the best prediction corresponding to a large number of samples, other damage levels also demonstrated a range of occurrence probabilities of different damage levels at the same tsunami parameters. These criteria made it easy to comprehend the probability of exceeding certain critical loss ratio ranges, as opposed to a measure of average loss, which requires more user knowledge to make an assessment and understand the implications.

Surveyed maximum tsunami heights were obtained by the 2011 Tohoku Earthquake Tsunami Joint Survey Group (2011). One maximum tsunami height along the coast was selected as a representative value for each port. The selected maximum tsunami height was then applied to all vessels in the ports to which they were registered, because the exact locations of the vessels in the ports were unknown. As a result, the use of the maximum tsunami height for each port was conservative, and there might be a slight overestimation because it was possible that not all vessel damage corresponded to the same maximum tsunami height in the port. This was a limitation because the positions of the vessels were dynamic, and it was not possible to know the exact locations of all vessels.

The 2011 Japan tsunami was reconstructed based on the source model developed by Sugino et al. (2013) to determine the maximum tsunami flow velocity. They divided the predicted source area into 48 segments with dimensions of $50\times 30\hspace{0.17em}\text{km}$ for segments near the trench line and $50\times 50\hspace{0.17em}\text{km}$ for the other segments. The predicted amount of slip in each segment varied from 0.0 m to a maximum of 77.9 m near the trench. Based on the preceding fault parameters, the initial sea surface condition was assumed to be the same as the sea bottom deformation, which was calculated using the equations proposed by Okada (1985). Nonlinear shallow water equations, as described in Imamura (1996), were used in this study to simulate the tsunami inundation.

In this study, the simulation of the 2011 Japan tsunami was performed for two domains of bathymetric data obtained from the Council of Disaster Prevention of Japan to construct a nested grid system with cells $405\times 135\hspace{0.17em}\text{m}$ in size. The existence of breakwaters and other structures was not taken into account in the simulation because of the size selected for the numerical grid for the smallest domain (135 m). The maximum tsunami height and flow velocity results from the simulation are shown in Fig. 5. The modeled tsunami heights were verified using survey data obtained by the 2011 Tohoku Earthquake Tsunami Joint Survey Group (2011). The verification was based on $K$ and $\kappa$, the Aida numbers (Aida 1978), defined as follows:In these equations, $x_i$ and $y_i$ = surveyed and computed tsunami heights at location $i$; and $n$ = total number of data points. Thus, $K$ is defined as the geometric mean of $K_i$, and $\kappa$ is defined as deviation or variance from $K$. These indexes are used as criteria to validate the model through comparisons between modeled and measured tsunamis. The Japan Society of Civil Engineers (JSCE 2002) provides guidelines that suggest that $0.95<K<1.05$ and $\kappa <1.45$ indicate good agreement between the tsunami source model and the propagation and/or inundation model evaluation. Values of $K=0.96$ and $\kappa =1.43$ were obtained for $n=\mathrm{1,352}$, which indicates fairly good agreement between the simulated tsunami height and the observed data. The simulated flow velocity was verified using the tsunami flow velocity that was estimated using available survivor videos from the Sendai plain (Hayashi and Koshimura 2013) and Kesennuma City (Fritz et al. 2012). Because the smallest grid size used in the numerical model was 135 m, it was not possible to conduct a point-by-point verification. Therefore, a square of $500\times 500\hspace{0.17em}\text{m}$ in size was defined at each place where the tsunami velocity was observed in a survivor video, and the range of the maximum simulated tsunami velocities was compared with the range of velocities in the observed data. In Kesennuma City, the simulated tsunami velocity had maximum values in the range of $7.3\hspace{0.17em}\text{to}\hspace{0.17em}9.6\hspace{0.17em}\text{m}/\text{s}$ (Fig. 6), whereas the data from the survivor videos (Fritz et al. 2012) suggested a range from $3\hspace{0.17em}\text{to}\hspace{0.17em}11\hspace{0.17em}\text{m}/\text{s}$. For the Sendai plain, a range of maximum simulated velocities of $5.7\hspace{0.17em}\text{to}\hspace{0.17em}6.5,4.7\hspace{0.17em}\text{to}\hspace{0.17em}5.1,\text{and}\hspace{0.17em}4.2\hspace{0.17em}\text{to}\hspace{0.17em}4.6\hspace{0.17em}\text{m}/\text{s}$ was obtained for squares a, b, and c, respectively, in Fig. (7). Hayashi and Koshimura (2013) reported a range from $5.5\hspace{0.17em}\text{to}\hspace{0.17em}8\hspace{0.17em}\text{m}/\text{s}$ for square a, a value of $3.5\hspace{0.17em}\text{m}/\text{s}$ for square b, and a value of $3.5\hspace{0.17em}\text{m}/\text{s}$ for square c. This comparison suggested that the simulated tsunami flow velocities were comparable to the velocity values determined from the survivor videos.

The values of the maximum tsunami flow velocity at each port was then extracted. The flow velocity values were then used, together with the damage data for boats in the ports, to develop loss functions for the vessels.

The vessel damage probabilities for each damage ratio were calculated and plotted against a median value within the range of the selected samples (varied from many tens to many hundreds). Linear regression analysis was performed to develop the loss functions. Assuming a log-normal distribution of the response, the cumulative probability $P$ of occurrence of damage is given by either Eq. (5) or Eq. (6)where $\mathrm{\Phi }$ = standard normal distribution function; $x$ = hydrodynamic features of the tsunami, used as demand parameters (e.g., inundation depth, current velocity, and hydrodynamic force); and $\mu$ and $\sigma$ (or ${\mu }^{\prime }$ and ${\sigma }^{\prime }$) = the mean and standard deviation, respectively, of $x$ (or $\ln x$). The two statistical parameters of the loss function, $\mu$ and $\sigma$ (or ${\mu }^{\prime }$ and ${\sigma }^{\prime }$), are obtained by plotting $x$ (or $\ln x$) against the inverse of $\mathrm{\Phi }$ on normal or log-normal probability paper and performing least-squares fitting of this plot. Consequently, two parameters are obtained by determining the intercept ($=\mu \hspace{0.17em}\text{or}\hspace{0.17em}{\mu }^{\prime }$) and the slope ($=\sigma \hspace{0.17em}\text{or}\hspace{0.17em}{\sigma }^{\prime }$) in Eq. (7) or Eq. (8)Through the regression analysis, the parameter values that yielded the best fit (in terms of minimizing the sum of the squared errors) of the loss functions to the inundation depths were determined.

The results of this study (Fig. 8) were compared with those of previous studies using the same threshold tsunami heights of 2, 5, and 10 m. For vessels weighing 5 t or less with outboard motors, the damage probabilities corresponding to the three tsunami heights were 0.4, 0.7, and 0.9, respectively. The damage probabilities for vessels weighing 5 t or less with inboard motors were 0.2, 0.5, and 0.75, respectively. As expected, vessels weighing 5–20 t had the least damage; their damage probabilities decreased to 0.05, 0.2, and 0.45, respectively. These results were consistent with those of previous studies in which vessels weighing 5–20 t had less damage because of the resistance of their weight to wave motion and because of the strength of their structures. The construction materials were also important; vessels weighing 5 t or less were typically made of FRP or occasionally wood, both of which are weaker than the steel and aluminum that are commonly used in larger vessels. Moreover, once FRP vessels were damaged, it is quite difficult to repair them, because they cannot be partially repaired as steel vessels can. In addition, the influence of motor type was confirmed. For the same tsunami height, vessels with outboard motors had a higher probability of damage because an outboard motor was more likely to be damaged or destroyed than an inboard motor, because of its location. The distance from the tsunami source was also found to influence the probability of damage; the curves for locations near the source increased more rapidly because these locations were attacked by higher tsunamis and left less time for fishermen to respond (Fig. 9). For example, the damage probabilities for a 2-m tsunami height increased to 0.7, 0.5, and 0.1 for the three types of vessels considered by locations. In contrast, for a location far from the tsunami source (Fig. 10), the damage probabilities for a 2-m tsunami height decreased to 0.2, 0.1, and 0.03, respectively. That is, locations farther from the tsunami source had lower damage probabilities because they were hit by lower tsunami heights and slower tsunami arrival times, which allowed more opportunity for vessels to evacuate to deep sea.

The results for the whole region are shown in Fig. 11. For vessels weighing 5 t or less with outboard motors, for a threshold flow velocity of $1\hspace{0.17em}\text{m}/\text{s}$, as mentioned previously, the probability of damage level 10 was found to be approximately 0.6, whereas the probability of damage level 2 was approximately 0.9. The probabilities of all types of damage levels approached 1.0 when the flow velocity was more than $5\hspace{0.17em}\text{m}/\text{s}$. For vessels weighing 5 t or less with inboard motors, at $1\hspace{0.17em}\text{m}/\text{s}$, the damage probability for level 10 decreased to 0.4, whereas it increased to 0.8 when the flow velocity was $5\hspace{0.17em}\text{m}/\text{s}$. For vessels weighing 5–20 t, the damage probability for level 10 at a flow velocity of $1\hspace{0.17em}\text{m}/\text{s}$ decreased to 0.1, but it was approximately 0.5 when the velocity was $5\hspace{0.17em}\text{m}/\text{s}$. These results confirmed the influence of tonnage and motor type on the vulnerability of vessels to tsunami damage reported in the previous section on tsunami height. The effect of distance from the tsunami source (Figs. 12 and 13) was similar to that of tsunami height, for the same reasons as mentioned in the previous section.